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People matching "Complex analysis"

Associate Professor Sanjeeva Balasuriya
Senior Lecturer in Applied Mathematics


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Dr David Baraglia
ARC DECRA Fellow, APD Fellow


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Associate Professor Nicholas Buchdahl
Reader in Pure Mathematics


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Associate Professor Gary Glonek
Associate Professor in Statistics


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Associate Professor Inge Koch
Associate Professor in Statistics


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Professor Finnur Larusson
Associate Professor in Pure Mathematics


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Professor Tony Roberts
Professor of Applied Mathematics


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Professor Patty Solomon
Professor of Statistical Bioinformatics


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Dr Simon Tuke
Lecturer in Statistics


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Courses matching "Complex analysis"

Analysis of multivariable and high dimensional data

Multivariate analysis of data is performed with the aims to 1. understand the structure in data and summarise the data in simpler ways; 2. understand the relationship of one part of the data to another part; and 3. make decisions or draw inferences based on data. The statistical analyses of multivariate data extend those of univariate data, and in doing so require more advanced mathematical theory and computational techniques. The course begins with a discussion of the three classical methods Principal Component Analysis, Canonical Correlation Analysis and Discriminant Analysis which correspond to the aims above. We also learn about Cluster Analysis, Factor Analysis and newer methods including Independent Component Analysis. For most real data the underlying distribution is not known, but if the assumptions of multivariate normality of the data hold, extra properties can be derived. Our treatment combines ideas, theoretical properties and a strong computational component for each of the different methods we discuss. For the computational part -- with Matlab -- we make use of real data and learn the use of simulations in order to assess the performance of different methods in practice. Topics covered: 1. Introduction to multivariate data, the multivariate normal distribution 2. Principal Component Analysis, theory and practice 3. Canonical Correlation Analysis, theory and practice 4. Discriminant Analysis, Fisher's LDA, linear and quadratic DA 5. Cluster Analysis: hierarchical and k-means methods 6. Factor Analysis and latent variables 7. Independent Component Analysis including an Introduction to Information Theory The course will be based on my forthcoming monograph Analysis of Multivariate and High-Dimensional Data - Theory and Practice, to be published by Cambridge University Press.

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Complex Analysis III

When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex-)differentiable functions turn out to have many remarkable properties not enjoyed by their real analogues. These functions, usually known as holomorphic functions, have numerous applications in areas such as engineering, physics, differential equations and number theory, to name just a few. The focus of this course is on the study of holomorphic functions and their most important basic properties. Topics covered are: Complex numbers and functions; complex limits and differentiability; elementary examples; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouche's Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues; outlines of the Jordan Curve Theorem, Montel's Theorem and the Riemann Mapping Theorem.

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Integration and Analysis III

The Riemann integral works well for continuous functions on closed bounded intervals, but it has certain deficiencies that cause problems, for example, in Fourier analysis and in the theory of differential equations. To overcome such deficiencies, a "new and improved" version of the integral was developed around the beginning of the twentieth century, and it is this theory with which this course is concerned. The underlying basis of the theory, measure theory, has important applications not just in analysis but also in the modern theory of probability. Topics covered are: Set theory; Lebesgue outer measure; measurable sets; measurable functions. Integration of measurable functions over measurable sets. Convergence of sequences of functions and their integrals. General measure spaces and product measures. Fubini and Tonelli's theorems. Lp spaces. The Radon-Nikodym theorem. The Riesz representation theorem. Integration and Differentiation.

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Multivariable and Complex Calculus

The mathematics required to describe most "real life" systems involves functions of more than one variable, so the differential and integral calculus developed in a first course in Calculus must be extended to functions of more variables. In this course, the key results of one-variable calculus are extended to higher dimensions: differentiation, integration, and the link between them provided by the Fundamental Theorem of Calculus are all generalised. The machinery developed can be applied to another generalisation of one-variable Calculus, namely to complex calculus, and the course also provides an introduction to this subject. The material covered in this course forms the basis for mathematical analysis and application across an extremely broad range of areas, essential for anyone studying the hard sciences, engineering, or mathematical economics/finance. Topics covered are: introduction to multivariable calculus; differentiation of scalar- and vector-valued functions; higher-order derivatives, extrema, Lagrange multipliers and the implicit function theorem; integration over regions, volumes, paths and surfaces; Green's, Stokes' and Gauss's theorems; differential forms; curvilinear coordinates; an introduction to complex numbers and functions; complex differentiation; complex integration and Cauchy's theorems; and conformal mappings.

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Real Analysis

Modern mathematics and physics rely on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called ``analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations.

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Statistical Analysis and Modelling 1

This is a first course in Statistics for mathematically inclined students. It will address the key principles underlying commonly used statistical methods such as confidence intervals, hypothesis tests, inference for means and proportions, and linear regression. It will develop a deeper mathematical understanding of these ideas, many of which will be familiar from studies in secondary school. The application of basic and more advanced statistical methods will be illustrated on a range of problems from areas such as medicine, science, technology, government, commerce and manufacturing. The use of the statistical package SPSS will be developed through a sequence of computer practicals. Topics covered will include: basic probability and random variables, fundamental distributions, inference for means and proportions, comparison of independent and paired samples, simple linear regression, diagnostics and model checking, multiple linear regression, simple factorial models, models with factors and continuous predictors.

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Topics in analysis

Description TBA

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Topology and Analysis III

Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance. Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.

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Events matching "Complex analysis"

Stability of time-periodic flows
15:10 Fri 10 Mar, 2006 :: G08 Mathematics Building University of Adelaide :: Prof. Andrew Bassom, School of Mathematics and Statistics, University of Western Australia

Time-periodic shear layers occur naturally in a wide range of applications from engineering to physiology. Transition to turbulence in such flows is of practical interest and there have been several papers dealing with the stability of flows composed of a steady component plus an oscillatory part with zero mean. In such flows a possible instability mechanism is associated with the mean component so that the stability of the flow can be examined using some sort of perturbation-type analysis. This strategy fails when the mean part of the flow is small compared with the oscillatory component which, of course, includes the case when the mean part is precisely zero.

This difficulty with analytical studies has meant that the stability of purely oscillatory flows has relied on various numerical methods. Until very recently such techniques have only ever predicted that the flow is stable, even though experiments suggest that they do become unstable at high enough speeds. In this talk I shall expand on this discrepancy with emphasis on the particular case of the so-called flat Stokes layer. This flow, which is generated in a deep layer of incompressible fluid lying above a flat plate which is oscillated in its own plane, represents one of the few exact solutions of the Navier-Stokes equations. We show theoretically that the flow does become unstable to waves which propagate relative to the basic motion although the theory predicts that this occurs much later than has been found in experiments. Reasons for this discrepancy are examined by reference to calculations for oscillatory flows in pipes and channels. Finally, we propose some new experiments that might reduce this disagreement between the theoretical predictions of instability and practical realisations of breakdown in oscillatory flows.
Homological algebra and applications - a historical survey
15:10 Fri 19 May, 2006 :: G08 Mathematics Building University of Adelaide :: Prof. Amnon Neeman

Homological algebra is a curious branch of mathematics; it is a powerful tool which has been used in many diverse places, without any clear understanding why it should be so useful. We will give a list of applications, proceeding chronologically: first to topology, then to complex analysis, then to algebraic geometry, then to commutative algebra and finally (if we have time) to non-commutative algebra. At the end of the talk I hope to be able to say something about the part of homological algebra on which I have worked, and its applications. That part is derived categories.
A Bivariate Zero-inflated Poisson Regression Model and application to some Dental Epidemiological data
14:10 Fri 27 Oct, 2006 :: G08 Mathematics Building University of Adelaide :: University Prof Sudhir Paul

Data in the form of paired (pre-treatment, post-treatment) counts arise in the study of the effects of several treatments after accounting for possible covariate effects. An example of such a data set comes from a dental epidemiological study in Belo Horizonte (the Belo Horizonte caries prevention study) which evaluated various programmes for reducing caries. Also, these data may show extra pairs of zeros than can be accounted for by a simpler model, such as, a bivariate Poisson regression model. In such situations we propose to use a zero-inflated bivariate Poisson regression (ZIBPR) model for the paired (pre-treatment, posttreatment) count data. We develop EM algorithm to obtain maximum likelihood estimates of the parameters of the ZIBPR model. Further, we obtain exact Fisher information matrix of the maximum likelihood estimates of the parameters of the ZIBPR model and develop a procedure for testing treatment effects. The procedure to detect treatment effects based on the ZIBPR model is compared, in terms of size, by simulations, with an earlier procedure using a zero-inflated Poisson regression (ZIPR) model of the post-treatment count with the pre-treatment count treated as a covariate. The procedure based on the ZIBPR model holds level most effectively. A further simulation study indicates good power property of the procedure based on the ZIBPR model. We then compare our analysis, of the decayed, missing and filled teeth (DMFT) index data from the caries prevention study, based on the ZIBPR model with the analysis using a zero-inflated Poisson regression model in which the pre-treatment DMFT index is taken to be a covariate
Statistical convergence of sequences of complex numbers with application to Fourier series
15:10 Tue 27 Mar, 2007 :: G08 Mathematics Building University of Adelaide :: Prof. Ferenc Morics

Media...
The concept of statistical convergence was introduced by Henry Fast and Hugo Steinhaus in 1951. But in fact, it was Antoni Zygmund who first proved theorems on the statistical convergence of Fourier series, using the term \"almost convergence\". A sequence $\\{x_k : k=1,2\\ldots\\}$ of complex numbers is said to be statistically convergent to $\\xi$ if for every $\\varepsilon >0$ we have $$\\lim_{n\\to \\infty} n^{-1} |\\{1\\le k\\le n: |x_k-\\xi| > \\varepsilon\\}| = 0.$$ We present the basic properties of statistical convergence, and extend it to multiple sequences. We also discuss the convergence behavior of Fourier series.
Identifying the source of photographic images by analysis of JPEG quantization artifacts
15:10 Fri 27 Apr, 2007 :: G08 Mathematics Building University of Adelaide :: Dr Matthew Sorell

Media...
In a forensic context, digital photographs are becoming more common as sources of evidence in criminal and civil matters. Questions that arise include identifying the make and model of a camera to assist in the gathering of physical evidence; matching photographs to a particular camera through the camera’s unique characteristics; and determining the integrity of a digital image, including whether the image contains steganographic information. From a digital file perspective, there is also the question of whether metadata has been deliberately modified to mislead the investigator, and in the case of multiple images, whether a timeline can be established from the various timestamps within the file, imposed by the operating system or determined by other image characteristics. This talk is concerned specifically with techniques to identify the make, model series and particular source camera model given a digital image. We exploit particular characteristics of the camera’s JPEG coder to demonstrate that such identification is possible, and that even when an image has subsequently been re-processed, there are often sufficient residual characteristics of the original coding to at least narrow down the possible camera models of interest.
The Linear Algebra of Internet Search Engines
15:10 Fri 5 Oct, 2007 :: G04 Napier Building University of Adelaide :: Dr Lesley Ward :: School of Mathematics and Statistics, University of South Australia

We often want to search the web for information on a given topic. Early web-search algorithms worked by counting up the number of times the words in a query topic appeared on each webpage. If the topic words appeared often on a given page, that page was ranked highly as a source of information on that topic. More recent algorithms rely on Link Analysis. People make judgments about how useful a given page is for a given topic, and they express these judgments through the hyperlinks they choose to put on their own webpages. Link-analysis algorithms aim to mine the collective wisdom encoded in the resulting network of links. I will discuss the linear algebra that forms the common underpinning of three link-analysis algorithms for web search. I will also present some work on refining one such algorithm, Kleinberg's HITS algorithm. This is joint work with Joel Miller, Greg Rae, Fred Schaefer, Ayman Farahat, Tom LoFaro, Tracy Powell, Estelle Basor, and Kent Morrison. It originated in a Mathematics Clinic project at Harvey Mudd College.
Rubber Ballons -- Prototypes of Hysteresis
15:10 Fri 16 Nov, 2007 :: G04 Napier Building University of Adelaide :: Emeritus Prof. Ingo Muller :: Technical University Berlin

Rubber balloons are characterized by a non-monotone pressure-radius relation which presages interesting non-trivial stability problems. A stability criterion is developed and exploited in order to show that the balloon may be stabilized at any radius by loading it with a piston under an elastic spring, if only the spring is hard enough. If two connected balloons are subject to an inflation-deflation cycle, the pressure-radius curve exhibits a fairly simple hysteresis loop. More complex hysteresis loops appear when more balloons are all inflated together. And if many balloons are inflated and deflated at the same time, the hysteresis loop assumes the form reminiscent of pseudo-elasticity. Stability in those complex cases is determined by a simple suggestive argument. References: [1] W.Kitsche, I.Muller, P.Strehlow. Simulation of pseudo-elastic behaviour in a system of rubber balloons. In: Metastability and Incompletely Posed Problems, S.Antman, J.L.Ericksen, D.Kinderlehrer, I.Muller (eds.) IMA Volume No.3, Springer Verlag, New York (1987) [2] I.Muller, P.Strehlow, Rubber and Rubber Balloons, Springer Lecture Notes on Physics, Springer Verlag, Heidelberg (2004)
Computational Methods for Phase Response Analysis of Circadian Clocks
15:10 Fri 18 Jul, 2008 :: G04 Napier Building University of Adelaide. :: Prof. Linda Petzold :: Dept. of Mechanical and Environmental Engineering, University of California, Santa Barbara

Circadian clocks govern daily behaviors of organisms in all kingdoms of life. In mammals, the master clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. It is composed of thousands of neurons, each of which contains a sloppy oscillator - a molecular clock governed by a transcriptional feedback network. Via intercellular signaling, the cell population synchronizes spontaneously, forming a coherent oscillation. This multi-oscillator is then entrained to its environment by the daily light/dark cycle.

Both at the cellular and tissular levels, the most important feature of the clock is its ability not simply to keep time, but to adjust its time, or phase, to signals. We present the parametric impulse phase response curve (pIPRC), an analytical analog to the phase response curve (PRC) used experimentally. We use the pIPRC to understand both the consequences of intercellular signaling and the light entrainment process. Further, we determine which model components determine the phase response behavior of a single oscillator by using a novel model reduction technique. We reduce the number of model components while preserving the pIPRC and then incorporate the resultant model into a couple SCN tissue model. Emergent properties, including the ability of the population to synchronize spontaneously are preserved in the reduction. Finally, we present some mathematical tools for the study of synchronization in a network of coupled, noisy oscillators.

Betti's Reciprocal Theorem for Inclusion and Contact Problems
15:10 Fri 1 Aug, 2008 :: G03 Napier Building University of Adelaide :: Prof. Patrick Selvadurai :: Department of Civil Engineering and Applied Mechanics, McGill University

Enrico Betti (1823-1892) is recognized in the mathematics community for his pioneering contributions to topology. An equally important contribution is his formulation of the reciprocity theorem applicable to elastic bodies that satisfy the classical equations of linear elasticity. Although James Clerk Maxwell (1831-1879) proposed a law of reciprocal displacements and rotations in 1864, the contribution of Betti is acknowledged for its underlying formal mathematical basis and generality. The purpose of this lecture is to illustrate how Betti's reciprocal theorem can be used to full advantage to develop compact analytical results for certain contact and inclusion problems in the classical theory of elasticity. Inclusion problems are encountered in number of areas in applied mechanics ranging from composite materials to geomechanics. In composite materials, the inclusion represents an inhomogeneity that is introduced to increase either the strength or the deformability characteristics of resulting material. In geomechanics, the inclusion represents a constructed material region, such as a ground anchor, that is introduced to provide load transfer from structural systems. Similarly, contact problems have applications to the modelling of the behaviour of indentors used in materials testing to the study of foundations used to distribute loads transmitted from structures. In the study of conventional problems the inclusions and the contact regions are directly loaded and this makes their analysis quite straightforward. When the interaction is induced by loads that are placed exterior to the indentor or inclusion, the direct analysis of the problem becomes inordinately complicated both in terns of formulation of the integral equations and their numerical solution. It is shown by a set of selected examples that the application of Betti's reciprocal theorem leads to the development of exact closed form solutions to what would otherwise be approximate solutions achievable only through the numerical solution of a set of coupled integral equations.
The Role of Walls in Chaotic Mixing
15:10 Fri 22 Aug, 2008 :: G03 Napier Building University of Adelaide :: Dr Jean-Luc Thiffeault :: Department of Mathematics, University of Wisconsin - Madison

I will report on experiments of chaotic mixing in closed and open vessels, in which a highly viscous fluid is stirred by a moving rod. In these experiments we analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and observe a slow algebraic decay, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel wall, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1-D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results. I also discuss possible ways of avoiding the limiting role of walls.

This is joint work with Emmanuelle Gouillart, Olivier Dauchot, and Stephane Roux.

Probabilistic models of human cognition
15:10 Fri 29 Aug, 2008 :: G03 Napier Building University of Adelaide :: Dr Daniel Navarro :: School of Psychology, University of Adelaide

Over the last 15 years a fairly substantial psychological literature has developed in which human reasoning and decision-making is viewed as the solution to a variety of statistical problems posed by the environments in which we operate. In this talk, I briefly outline the general approach to cognitive modelling that is adopted in this literature, which relies heavily on Bayesian statistics, and introduce a little of the current research in this field. In particular, I will discuss work by myself and others on the statistical basis of how people make simple inductive leaps and generalisations, and the links between these generalisations and how people acquire word meanings and learn new concepts. If time permits, the extensions of the work in which complex concepts may be characterised with the aid of nonparametric Bayesian tools such as Dirichlet processes will be briefly mentioned.
Free surface Stokes flows with surface tension
15:10 Fri 5 Sep, 2008 :: G03 Napier Building University of Adelaide :: Prof. Darren Crowdy :: Imperial College London

In this talk, we will survey a number of different free boundary problems involving slow viscous (Stokes) flows in which surface tension is active on the free boundary. Both steady and unsteady flows will be considered. Motivating applications range from industrial processes such as viscous sintering (where end-products are formed as a result of the surface-tension-driven densification of a compact of smaller particles that are heated in order that they coalesce) to biological phenomena such as understanding how organisms swim (i.e. propel themselves) at low Reynolds numbers. Common to our approach to all these problems will be an analytical/theoretical treatment of model problems via complex variable methods -- techniques well-known at infinite Reynolds numbers but used much less often in the Stokes regime. These model problems can give helpful insights into the behaviour of the true physical systems.
Mathematical modelling of blood flow in curved arteries
15:10 Fri 12 Sep, 2008 :: G03 Napier Building University of Adelaide :: Dr Jennifer Siggers :: Imperial College London

Atherosclerosis, characterised by plaques, is the most common arterial disease. Plaques tend to develop in regions of low mean wall shear stress, and regions where the wall shear stress changes direction during the course of the cardiac cycle. To investigate the effect of the arterial geometry and driving pressure gradient on the wall shear stress distribution we consider an idealised model of a curved artery with uniform curvature. We assume that the flow is fully-developed and seek solutions of the governing equations, finding the effect of the parameters on the flow and wall shear stress distribution. Most previous work assumes the curvature ratio is asymptotically small; however, many arteries have significant curvature (e.g. the aortic arch has curvature ratio approx 0.25), and in this work we consider in particular the effect of finite curvature.

We present an extensive analysis of curved-pipe flow driven by a steady and unsteady pressure gradients. Increasing the curvature causes the shear stress on the inside of the bend to rise, indicating that the risk of plaque development would be overestimated by considering only the weak curvature limit.

Oceanographic Research at the South Australian Research and Development Institute: opportunities for collaborative research
15:10 Fri 21 Nov, 2008 :: Napier G04 :: Associate Prof John Middleton :: South Australian Research and Development Institute

Increasing threats to S.A.'s fisheries and marine environment have underlined the increasing need for soundly based research into the ocean circulation and ecosystems (phyto/zooplankton) of the shelf and gulfs. With support of Marine Innovation SA, the Oceanography Program has within 2 years, grown to include 6 FTEs and a budget of over $4.8M. The program currently leads two major research projects, both of which involve numerical and applied mathematical modelling of oceanic flow and ecosystems as well as statistical techniques for the analysis of data. The first is the implementation of the Southern Australian Integrated Marine Observing System (SAIMOS) that is providing data to understand the dynamics of shelf boundary currents, monitor for climate change and understand the phyto/zooplankton ecosystems that under-pin SA's wild fisheries and aquaculture. SAIMOS involves the use of ship-based sampling, the deployment of underwater marine moorings, underwater gliders, HF Ocean RADAR, acoustic tracking of tagged fish and Autonomous Underwater vehicles.

The second major project involves measuring and modelling the ocean circulation and biological systems within Spencer Gulf and the impact on prawn larval dispersal and on the sustainability of existing and proposed aquaculture sites. The discussion will focus on opportunities for collaborative research with both faculty and students in this exciting growth area of S.A. science.

Bursts and canards in a pituitary lactotroph model
15:10 Fri 6 Mar, 2009 :: Napier LG29 :: Dr Martin Wechselberger :: University of Sydney

Bursting oscillations in nerve cells have been the focus of a great deal of attention by mathematicians. These are typically studied by taking advantage of multiple time-scales in the system under study to perform a singular perturbation analysis. Bursting also occurs in hormone-secreting pituitary cells, but is characterized by fast bursts with small electrical impulses. Although the separation of time-scales is not as clear, singular perturbation analysis is still the key to understand the bursting mechanism. In particular, we will show that canards are responsible for the observed oscillatory behaviour.
Geometric analysis on the noncommutative torus
13:10 Fri 20 Mar, 2009 :: School Board Room :: Prof Jonathan Rosenberg :: University of Maryland

Noncommutative geometry (in the sense of Alain Connes) involves replacing a conventional space by a "space" in which the algebra of functions is noncommutative. The simplest truly non-trivial noncommutative manifold is the noncommutative 2-torus, whose algebra of functions is also called the irrational rotation algebra. I will discuss a number of recent results on geometric analysis on the noncommutative torus, including the study of nonlinear noncommutative elliptic PDEs (such as the noncommutative harmonic map equation) and noncommutative complex analysis (with noncommutative elliptic functions).
Understanding optimal linear transient growth in complex-geometry flows
15:00 Fri 27 Mar, 2009 :: Napier LG29 :: Associate Prof Hugh Blackburn :: Monash University

Classification and compact complex manifolds I
13:10 Fri 17 Apr, 2009 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide

Classification and compact complex manifolds II
13:10 Fri 24 Apr, 2009 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide

Four classes of complex manifolds
13:10 Fri 8 May, 2009 :: School Board Room :: A/Prof Finnur Larusson :: University of Adelaide

We introduce the four classes of complex manifolds defined by having few or many holomorphic maps to or from the complex plane. Two of these classes have played an important role in complex geometry for a long time. A third turns out to be too large to be of much interest. The fourth class has only recently emerged from work of Abel Prize winner Mikhail Gromov.
Lagrangian fibrations on holomorphic symplectic manifolds I: Holomorphic Lagrangian fibrations
13:10 Fri 5 Jun, 2009 :: School Board Room :: Dr Justin Sawon :: Colorado State University

A compact K{\"a}hler manifold $X$ is a holomorphic symplectic manifold if it admits a non-degenerate holomorphic two-form $\sigma$. According to a theorem of Matsushita, fibrations on $X$ must be of a very restricted type: the fibres must be Lagrangian with respect to $\sigma$ and the generic fibre must be a complex torus. Moreover, it is expected that the base of the fibration must be complex projective space, and this has been proved by Hwang when $X$ is projective. The simplest example of these {\em Lagrangian fibrations\/} are elliptic K3 surfaces. In this talk we will explain the role of elliptic K3s in the classification of K3 surfaces, and the (conjectural) generalization to higher dimensions.
Statistical analysis for harmonized development of systemic organs in human fetuses
11:00 Thu 17 Sep, 2009 :: School Board Room :: Prof Kanta Naito :: Shimane University

The growth processes of human babies have been studied sufficiently in scientific fields, but there have still been many issues about the developments of human fetus which are not clarified. The aim of this research is to investigate the developing process of systemic organs of human fetuses based on the data set of measurements of fetus's bodies and organs. Specifically, this talk is concerned with giving a mathematical understanding for the harmonized developments of the organs of human fetuses. The method to evaluate such harmonies is proposed by the use of the maximal dilatation appeared in the theory of quasi-conformal mapping.
The proof of the Poincare conjecture
15:10 Fri 25 Sep, 2009 :: Napier 102 :: Prof Terrence Tao :: UCLA

In a series of three papers from 2002-2003, Grigori Perelman gave a spectacular proof of the Poincare Conjecture (every smooth compact simply connected three-dimensional manifold is topologically isomorphic to a sphere), one of the most famous open problems in mathematics (and one of the seven Clay Millennium Prize Problems worth a million dollars each), by developing several new groundbreaking advances in Hamilton's theory of Ricci flow on manifolds. In this talk I describe in broad detail how the proof proceeds, and briefly discuss some of the key turning points in the argument. About the speaker: Terence Tao was born in Adelaide, Australia, in 1975. He has been a professor of mathematics at UCLA since 1999, having completed his PhD under Elias Stein at Princeton in 1996. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Salem Prize in 2000, the Bochner Prize in 2002, the Fields Medal and SASTRA Ramanujan Prize in 2006, and the MacArthur Fellowship and Ostrowski Prize in 2007. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society and the Australian Academy of Sciences (Corresponding Member).
Modelling and pricing for portfolio credit derivatives
15:10 Fri 16 Oct, 2009 :: MacBeth Lecture Theatre :: Dr Ben Hambly :: University of Oxford

The current financial crisis has been in part precipitated by the growth of complex credit derivatives and their mispricing. This talk will discuss some of the background to the `credit crunch', as well as the models and methods used currently. We will then develop an alternative view of large basket credit derivatives, as functions of a stochastic partial differential equation, which addresses some of the shortcomings.
Analytic torsion for twisted de Rham complexes
13:10 Fri 30 Oct, 2009 :: School Board Room :: Prof Mathai Varghese :: University of Adelaide

We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.
Eigen-analysis of fluid-loaded compliant panels
15:10 Wed 9 Dec, 2009 :: Santos Lecture Theatre :: Prof Tony Lucey :: Curtin University of Technology

This presentation concerns the fluid-structure interaction (FSI) that occurs between a fluid flow and an arbitrarily deforming flexible boundary considered to be a flexible panel or a compliant coating that comprises the wetted surface of a marine vehicle. We develop and deploy an approach that is a hybrid of computational and theoretical techniques. The system studied is two-dimensional and linearised disturbances are assumed. Of particular novelty in the present work is the ability of our methods to extract a full set of fluid-structure eigenmodes for systems that have strong spatial inhomogeneity in the structure of the flexible wall.

We first present the approach and some results of the system in which an ideal, zero-pressure gradient, flow interacts with a flexible plate held at both its ends. We use a combination of boundary-element and finite-difference methods to express the FSI system as a single matrix equation in the interfacial variable. This is then couched in state-space form and standard methods used to extract the system eigenvalues. It is then shown how the incorporation of spatial inhomogeneity in the stiffness of the plate can be either stabilising or destabilising. We also show that adding a further restraint within the streamwise extent of a homogeneous panel can trigger an additional type of hydroelastic instability at low flow speeds. The mechanism for the fluid-to-structure energy transfer that underpins this instability can be explained in terms of the pressure-signal phase relative to that of the wall motion and the effect on this relationship of the added wall restraint.

We then show how the ideal-flow approach can be conceptually extended to include boundary-layer effects. The flow field is now modelled by the continuity equation and the linearised perturbation momentum equation written in velocity-velocity form. The near-wall flow field is spatially discretised into rectangular elements on an Eulerian grid and a variant of the discrete-vortex method is applied. The entire fluid-structure system can again be assembled as a linear system for a single set of unknowns - the flow-field vorticity and the wall displacements - that admits the extraction of eigenvalues. We then show how stability diagrams for the fully-coupled finite flow-structure system can be assembled, in doing so identifying classes of wall-based or fluid-based and spatio-temporal wave behaviour.

Critical sets of products of linear forms
13:10 Mon 14 Dec, 2009 :: School Board Room :: Dr Graham Denham :: University of Western Ontario, Canada

Suppose $f_1,f_2,\ldots,f_n$ are linear polynomials in $\ell$ variables and $\lambda_1,\lambda_2,\ldots,\lambda_n$ are nonzero complex numbers. The product $$ \Phi_\lambda=\Prod_{i=1}^n f_1^{\lambda_i}, $$ called a master function, defines a (multivalued) function on $\ell$-dimensional complex space, or more precisely, on the complement of a set of hyperplanes. Then it is easy to ask (but harder to answer) what the set of critical points of a master function looks like, in terms of some properties of the input polynomials and $\lambda_i$'s. In my talk I will describe the motivation for considering such a question. Then I will indicate how the geometry and combinatorics of hyperplane arrangements can be used to provide at least a partial answer.
Hartogs-type holomorphic extensions
13:10 Tue 15 Dec, 2009 :: School Board Room :: Prof Roman Dwilewicz :: Missouri University of Science and Technology

We will review holomorphic extension problems starting with the famous Hartogs extension theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent (partial) results of Al Boggess (Texas A&M Univ.), Zbigniew Slodkowski (Univ. Illinois at Chicago), and the speaker. The holomorphic extension problems for holomorphic or Cauchy-Riemann functions are fundamental problems in complex analysis of several variables. The talk will be very elementary, with many figures, and accessible to graduate and even advanced undergraduate students.
Group actions in complex geometry, I and II
13:10 Fri 8 Jan, 2010 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne

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Group actions in complex geometry, III and IV
10:10 Fri 15 Jan, 2010 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne

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Group actions in complex geometry, V and VI
10:10 Fri 22 Jan, 2010 :: School Board Room :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne

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Group actions in complex geometry, VII and VIII
10:10 Fri 29 Jan, 2010 :: Napier LG 23 :: Prof Frank Kutzschebauch, IGA Lecturer :: University of Berne

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Oka manifolds and Oka maps
13:10 Fri 29 Jan, 2010 :: Napier LG 23 :: Prof Franc Forstneric :: University of Ljubljana

In this survey lecture I will discuss a new class of complex manifolds and of holomorphic maps between them which I introduced in 2009 (F. Forstneric, Oka Manifolds, C. R. Acad. Sci. Paris, Ser. I, 347 (2009) 1017-1020). Roughly speaking, a complex manifold Y is said to be an Oka manifold if Y admits plenty of holomorphic maps from any Stein manifold (or Stein space) X to Y, in a certain precise sense. In particular, the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps must be a weak homotopy equivalence. One of the main results is that this class of manifolds can be characterized by a simple Runge approximation property for holomorphic maps from complex Euclidean spaces C^n to Y, with approximation on compact convex subsets of C^n. This answers in the affirmative a question posed by M. Gromov in 1989. I will also discuss the Oka properties of holomorphic maps and their characterization by approximation properties.
A solution to the Gromov-Vaserstein problem
15:10 Fri 29 Jan, 2010 :: Engineering North N 158 Chapman Lecture Theatre :: Prof Frank Kutzschebauch :: University of Berne, Switzerland

Any matrix in $SL_n (\mathbb C)$ can be written as a product of elementary matrices using the Gauss elimination process. If instead of the field of complex numbers, the entries in the matrix are elements of a more general ring, this becomes a delicate question. In particular, rings of complex-valued functions on a space are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in $m$ variables in case the size $n$ of the matrix is at least 3. In the topological category, the problem was solved by Thurston and Vaserstein. For holomorphic functions on $\mathbb C^m$, the problem was posed by Gromov in the 1980s. We report on a complete solution to Gromov's problem. A main tool is the Oka-Grauert-Gromov h-principle in complex analysis. Our main theorem can be formulated as follows: In the absence of obvious topological obstructions, the Gauss elimination process can be performed in a way that depends holomorphically on the matrix. This is joint work with Bj\"orn Ivarsson.
Proper holomorphic maps from strongly pseudoconvex domains to q-convex manifolds
13:10 Fri 5 Feb, 2010 :: School Board Room :: Prof Franc Forstneric :: University of Ljubljana

(Joint work with B. Drinovec Drnovsek, Amer. J. Math., in press.) I will discuss the existence of closed complex subvarieties of a complex manifold X that are proper holomorphic images of strongly pseudoconvex Stein domains. The main sufficient condition is expressed in terms of the Morse indices and of the number of positive Levi eigenvalues of an exhaustion function on X. Examples show that our condition cannot be weakened in general. I will describe optimal results for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to complex Euclidean spaces.
Holomorphic extension on complex spaces
14:10 Fri 5 Mar, 2010 :: School Board Room :: Prof Egmont Porten :: Mid Sweden University

Moduli spaces of stable holomorphic vector bundles II
13:10 Fri 30 Apr, 2010 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide

In this talk, I shall briefly review the notion of stability for holomorphic vector bundles on compact complex manifolds as discussed in the first part of this talk (28 August 2009). Then I shall attempt to compute some explicit examples in simple situations, illustrating the use of basic algebraic-geometric tools. The level of the talk will be appropriate for graduate students, particularly those who have been taking part in the algebraic geometry reading group meetings.
Estimation of sparse Bayesian networks using a score-based approach
15:10 Fri 30 Apr, 2010 :: School Board Room :: Dr Jessica Kasza :: University of Copenhagen

The estimation of Bayesian networks given high-dimensional data sets, with more variables than there are observations, has been the focus of much recent research. These structures provide a flexible framework for the representation of the conditional independence relationships of a set of variables, and can be particularly useful in the estimation of genetic regulatory networks given gene expression data.

In this talk, I will discuss some new research on learning sparse networks, that is, networks with many conditional independence restrictions, using a score-based approach. In the case of genetic regulatory networks, such sparsity reflects the view that each gene is regulated by relatively few other genes. The presented approach allows prior information about the overall sparsity of the underlying structure to be included in the analysis, as well as the incorporation of prior knowledge about the connectivity of individual nodes within the network.

Moduli spaces of stable holomorphic vector bundles III
13:10 Fri 14 May, 2010 :: School Board Room :: A/Prof Nicholas Buchdahl :: University of Adelaide

This talk is a continuation of the talk on 30 April. The same abstract applies: In this talk, I shall briefly review the notion of stability for holomorphic vector bundles on compact complex manifolds as discussed in the first part of this talk (28 August 2009). Then I shall attempt to compute some explicit examples in simple situations, illustrating the use of basic algebraic-geometric tools. The level of the talk will be appropriate for graduate students, particularly those who have been taking part in the algebraic geometry reading group meetings.
Understanding convergence of meshless methods: Vortex methods and smoothed particle hydrodynamics
15:10 Fri 14 May, 2010 :: Santos Lecture Theatre :: A/Prof Lou Rossi :: University of Delaware

Meshless methods such as vortex methods (VMs) and smoothed particle hydrodynamics (SPH) schemes offer many advantages in fluid flow computations. Particle-based computations naturally adapt to complex flow geometries and so provide a high degree of computational efficiency. Also, particle based methods avoid CFL conditions because flow quantities are integrated along characteristics. There are many approaches to improving numerical methods, but one of the most effective routes is quantifying the error through the direct estimate of residual quantities. Understanding the residual for particle schemes requires a different approach than for meshless schemes but the rewards are significant. In this seminar, I will outline a general approach to understanding convergence that has been effective in creating high spatial accuracy vortex methods, and then I will discuss some recent investigations in the accuracy of diffusion operators used in SPH computations. Finally, I will provide some sample Navier-Stokes computations of high Reynolds number flows using BlobFlow, an open source implementation of the high precision vortex method.
Whole genome analysis of repetitive DNA
15:10 Fri 21 May, 2010 :: Napier 209 :: Prof David Adelson :: University of Adelaide

The interspersed repeat content of mammalian genomes has been best characterized in human, mouse and cow. We carried out de novo identification of repeated elements in the equine genome and identified previously unknown elements present at low copy number. The equine genome contains typical eutherian mammal repeats. We analysed both interspersed and simple sequence repeats (SSR) genome-wide, finding that some repeat classes are spatially correlated with each other as well as with G+C content and gene density. Based on these spatial correlations, we have confirmed recently-described ancestral vs clade-specific genome territories defined by repeat content. Territories enriched for ancestral repeats tended to be contiguous domains. To determine if these territories were evolutionarily conserved, we compared these results with a similar analysis of the human genome, and observed similar ancestral repeat enriched domains. These results indicate that ancestral, evolutionarily conserved mammalian genome territories can be identified on the basis of repeat content alone. Interspersed repeats of different ages appear to be analogous to geologic strata, allowing identification of ancient vs newly remodelled regions of mammalian genomes.
Interpolation of complex data using spatio-temporal compressive sensing
13:00 Fri 28 May, 2010 :: Santos Lecture Theatre :: A/Prof Matthew Roughan :: School of Mathematical Sciences, University of Adelaide

Many complex datasets suffer from missing data, and interpolating these missing elements is a key task in data analysis. Moreover, it is often the case that we see only a linear combination of the desired measurements, not the measurements themselves. For instance, in network management, it is easy to count the traffic on a link, but harder to measure the end-to-end flows. Additionally, typical interpolation algorithms treat either the spatial, or the temporal components of data separately, but in many real datasets have strong spatio-temporal structure that we would like to exploit in reconstructing the missing data. In this talk I will describe a novel reconstruction algorithm that exploits concepts from the growing area of compressive sensing to solve all of these problems and more. The approach works so well on Internet traffic matrices that we can obtain a reasonable reconstruction with as much as 98% of the original data missing.
Vertex algebras and variational calculus I
13:10 Fri 4 Jun, 2010 :: School Board Room :: Dr Pedram Hekmati :: University of Adelaide

A basic operation in calculus of variations is the Euler-Lagrange variational derivative, whose kernel determines the extremals of functionals. There exists a natural resolution of this operator, called the variational complex. In this talk, I shall explain how to use tools from the theory of vertex algebras to explicitly construct the variational complex. This also provides a very convenient language for classifying and constructing integrable Hamiltonian evolution equations.
Vertex algebras and variational calculus II
13:10 Fri 11 Jun, 2010 :: School Board Room :: Dr Pedram Hekmati :: University of Adelaide

Last time I introduced the variational complex of an algebra of differential functions and gave a sketchy definition of a vertex algebra. This week I will make this notion more precise and explain how to apply it to the calculus of variations.
The mathematics of theoretical inference in cognitive psychology
15:10 Fri 11 Jun, 2010 :: Napier LG24 :: Prof John Dunn :: University of Adelaide

The aim of psychology in general, and of cognitive psychology in particular, is to construct theoretical accounts of mental processes based on observed changes in performance on one or more cognitive tasks. The fundamental problem faced by the researcher is that these mental processes are not directly observable but must be inferred from changes in performance between different experimental conditions. This inference is further complicated by the fact that performance measures may only be monotonically related to the underlying psychological constructs. State-trace analysis provides an approach to this problem which has gained increasing interest in recent years. In this talk, I explain state-trace analysis and discuss the set of mathematical issues that flow from it. Principal among these are the challenges of statistical inference and an unexpected connection to the mathematics of oriented matroids.
Some thoughts on wine production
15:05 Fri 18 Jun, 2010 :: School Board Room :: Prof Zbigniew Michalewicz :: School of Computer Science, University of Adelaide

In the modern information era, managers (e.g. winemakers) recognize the competitive opportunities represented by decision-support tools which can provide a significant cost savings & revenue increases for their businesses. Wineries make daily decisions on the processing of grapes, from harvest time (prediction of maturity of grapes, scheduling of equipment and labour, capacity planning, scheduling of crushers) through tank farm activities (planning and scheduling of wine and juice transfers on the tank farm) to packaging processes (bottling and storage activities). As such operation is quite complex, the whole area is loaded with interesting OR-related issues. These include the issues of global vs. local optimization, relationship between prediction and optimization, operating in dynamic environments, strategic vs. tactical optimization, and multi-objective optimization & trade-off analysis. During the talk we address the above issues; a few real-world applications will be shown and discussed to emphasize some of the presented material.
Eynard-Orantin invariants and enumerative geometry
13:10 Fri 6 Aug, 2010 :: Ingkarni Wardli B20 (Suite 4) :: Dr Paul Norbury :: University of Melbourne

As a tool for studying enumerative problems in geometry Eynard and Orantin associate multilinear differentials to any plane curve. Their work comes from matrix models but does not require matrix models (for understanding or calculations). In some sense they describe deformations of complex structures of a curve and conjectural relationships to deformations of Kahler structures of an associated object. I will give an introduction to their invariants via explicit examples, mainly to do with the moduli space of Riemann surfaces, in which the plane curve has genus zero.
Compound and constrained regression analyses for EIV models
15:05 Fri 27 Aug, 2010 :: Napier LG28 :: Prof Wei Zhu :: State University of New York at Stony Brook

In linear regression analysis, randomness often exists in the independent variables and the resulting models are referred to errors-in-variables (EIV) models. The existing general EIV modeling framework, the structural model approach, is parametric and dependent on the usually unknown underlying distributions. In this work, we introduce a general non-parametric EIV modeling framework, the compound regression analysis, featuring an intuitive geometric representation and a 1-1 correspondence to the structural model. Properties, examples and further generalizations of this new modeling approach are discussed in this talk.
Principal Component Analysis Revisited
15:10 Fri 15 Oct, 2010 :: Napier G04 :: Assoc. Prof Inge Koch :: University of Adelaide

Since the beginning of the 20th century, Principal Component Analysis (PCA) has been an important tool in the analysis of multivariate data. The principal components summarise data in fewer than the original number of variables without losing essential information, and thus allow a split of the data into signal and noise components. PCA is a linear method, based on elegant mathematical theory. The increasing complexity of data together with the emergence of fast computers in the later parts of the 20th century has led to a renaissance of PCA. The growing numbers of variables (in particular, high-dimensional low sample size problems), non-Gaussian data, and functional data (where the data are curves) are posing exciting challenges to statisticians, and have resulted in new research which extends the classical theory. I begin with the classical PCA methodology and illustrate the challenges presented by the complex data that we are now able to collect. The main part of the talk focuses on extensions of PCA: the duality of PCA and the Principal Coordinates of Multidimensional Scaling, Sparse PCA, and consistency results relating to principal components, as the dimension grows. We will also look at newer developments such as Principal Component Regression and Supervised PCA, nonlinear PCA and Functional PCA.
Real analytic sets in complex manifolds I: holomorphic closure dimension
13:10 Fri 4 Mar, 2011 :: Mawson 208 :: Dr Rasul Shafikov :: University of Western Ontario

After a quick introduction to real and complex analytic sets, I will discuss possible notions of complex dimension of real sets, and then discuss a structure theorem for the holomorphic closure dimension which is defined as the dimension of the smallest complex analytic germ containing the real germ.
Real analytic sets in complex manifolds II: complex dimension
13:10 Fri 11 Mar, 2011 :: Mawson 208 :: Dr Rasul Shafikov :: University of Western Ontario

Given a real analytic set R, denote by A the subset of R of points through which there is a nontrivial complex variety contained in R, i.e., A consists of points in R of positive complex dimension. I will discuss the structure of the set A.
Bioinspired computation in combinatorial optimization: algorithms and their computational complexity
15:10 Fri 11 Mar, 2011 :: 7.15 Ingkarni Wardli :: Dr Frank Neumann :: The University of Adelaide

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Bioinspired computation methods, such as evolutionary algorithms and ant colony optimization, are being applied successfully to complex engineering and combinatorial optimization problems. The computational complexity analysis of this type of algorithms has significantly increased the theoretical understanding of these successful algorithms. In this talk, I will give an introduction into this field of research and present some important results that we achieved for problems from combinatorial optimization. These results can also be found in my recent textbook "Bioinspired Computation in Combinatorial Optimization -- Algorithms and Their Computational Complexity".
Surface quotients of hyperbolic buildings
13:10 Fri 18 Mar, 2011 :: Mawson 208 :: Dr Anne Thomas :: University of Sydney

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons, and the link at each vertex is the complete bipartite graph K_{v,v}. We investigate and mostly determine the set of triples (p,v,g) for which there is a discrete group acting on I(p,v) so that the quotient is a compact orientable surface of genus g. Surprisingly, the existence of such a quotient depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. We use elementary group theory, combinatorics, algebraic topology and number theory. This is joint work with David Futer.
Classification for high-dimensional data
15:10 Fri 1 Apr, 2011 :: Conference Room Level 7 Ingkarni Wardli :: Associate Prof Inge Koch :: The University of Adelaide

For two-class classification problems Fisher's discriminant rule performs well in many scenarios provided the dimension, d, is much smaller than the sample size n. As the dimension increases, Fisher's rule may no longer be adequate, and can perform as poorly as random guessing. In this talk we look at new ways of overcoming this poor performance for high-dimensional data by suitably modifying Fisher's rule, and in particular we describe the 'Features Annealed Independence Rule (FAIR)? of Fan and Fan (2008) and a rule based on canonical correlation analysis. I describe some theoretical developments, and also show analysis of data which illustrate the performance of these modified rule.
Spherical tube hypersurfaces
13:10 Fri 8 Apr, 2011 :: Mawson 208 :: Prof Alexander Isaev :: Australian National University

We consider smooth real hypersurfaces in a complex vector space. Specifically, we are interested in tube hypersurfaces, i.e., hypersurfaces represented as the direct product of the imaginary part of the space and hypersurfaces lying in its real part. Tube hypersurfaces arise, for instance, as the boundaries of tube domains. The study of tube domains is a classical subject in several complex variables and complex geometry, which goes back to the beginning of the 20th century. Indeed, already Siegel found it convenient to realise certain symmetric domains as tubes. One can endow a tube hypersurface with a so-called CR-structure, which is the remnant of the complex structure on the ambient vector space. We impose on the CR-structure the condition of sphericity. One way to state this condition is to require a certain curvature (called the CR-curvature of the hypersurface) to vanish identically. Spherical tube hypersurfaces possess remarkable properties and are of interest from both the complex-geometric and affine-geometric points of view. I my talk I will give an overview of the theory of such hypersurfaces. In particular, I will mention an algebraic construction arising from this theory that has applications in abstract commutative algebra and singularity theory. I will speak about these applications in detail in my colloquium talk later today.
Algebraic hypersurfaces arising from Gorenstein algebras
15:10 Fri 8 Apr, 2011 :: 7.15 Ingkarni Wardli :: Associate Prof Alexander Isaev :: Australian National University

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To every Gorenstein algebra of finite dimension greater than 1 over a field of characteristic zero, and a projection on its maximal ideal with range equal to the annihilator of the ideal, one can associate a certain algebraic hypersurface lying in the ideal. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for the case of complex numbers leads to interesting consequences in singularity theory. Also, for the case of real numbers such hypersurfaces naturally arise in CR-geometry. In my talk I will discuss these hypersurfaces and some of their applications.
Centres of cyclotomic Hecke algebras
13:10 Fri 15 Apr, 2011 :: Mawson 208 :: A/Prof Andrew Francis :: University of Western Sydney

The cyclotomic Hecke algebras, or Ariki-Koike algebras $H(R,q)$, are deformations of the group algebras of certain complex reflection groups $G(r,1,n)$, and also are quotients of the ubiquitous affine Hecke algebra. The centre of the affine Hecke algebra has been understood since Bernstein in terms of the symmetric group action on the weight lattice. In this talk I will discuss the proof that over an arbitrary unital commutative ring $R$, the centre of the affine Hecke algebra maps \emph{onto} the centre of the cyclotomic Hecke algebra when $q-1$ is invertible in $R$. This is the analogue of the fact that the centre of the Hecke algebra of type $A$ is the set of symmetric polynomials in Jucys-Murphy elements (formerly known as he Dipper-James conjecture). Key components of the proof include the relationship between the trace functions on the affine Hecke algebra and on the cyclotomic Hecke algebra, and the link to the affine braid group. This is joint work with John Graham and Lenny Jones.
A strong Oka principle for embeddings of some planar domains into CxC*, I
13:10 Fri 6 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide

The Oka principle refers to a collection of results in complex analysis which state that there are only topological obstructions to solving certain holomorphically defined problems involving Stein manifolds. For example, a basic version of Gromov's Oka principle states that every continuous map from a Stein manifold into an elliptic complex manifold is homotopic to a holomorphic map. In these two talks I will discuss a new result showing that if we restrict the class of source manifolds to circular domains and fix the target as CxC* we can obtain a much stronger Oka principle: every continuous map from a circular domain S into CxC* is homotopic to a proper holomorphic embedding. This result has close links with the long-standing and difficult problem of finding proper holomorphic embeddings of Riemann surfaces into C^2, with additional motivation from other sources.
When statistics meets bioinformatics
12:10 Wed 11 May, 2011 :: Napier 210 :: Prof Patty Solomon :: School of Mathematical Sciences

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Bioinformatics is a new field of research which encompasses mathematics, computer science, biology, medicine and the physical sciences. It has arisen from the need to handle and analyse the vast amounts of data being generated by the new genomics technologies. The interface of these disciplines used to be information-poor, but is now information-mega-rich, and statistics plays a central role in processing this information and making it intelligible. In this talk, I will describe a published bioinformatics study which claimed to have developed a simple test for the early detection of ovarian cancer from a blood sample. The US Food and Drug Administration was on the verge of approving the test kits for market in 2004 when demonstrated flaws in the study design and analysis led to its withdrawal. We are still waiting for an effective early biomarker test for ovarian cancer.
A strong Oka principle for embeddings of some planar domains into CxC*, II
13:10 Fri 13 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide

The Oka principle refers to a collection of results in complex analysis which state that there are only topological obstructions to solving certain holomorphically defined problems involving Stein manifolds. For example, a basic version of Gromov's Oka principle states that every continuous map from a Stein manifold into an elliptic complex manifold is homotopic to a holomorphic map. In these two talks I will discuss a new result showing that if we restrict the class of source manifolds to circular domains and fix the target as CxC* we can obtain a much stronger Oka principle: every continuous map from a circular domain S into CxC* is homotopic to a proper holomorphic embedding. This result has close links with the long-standing and difficult problem of finding proper holomorphic embeddings of Riemann surfaces into C^2, with additional motivation from other sources.
Change detection in rainfall time series for Perth, Western Australia
12:10 Mon 16 May, 2011 :: 5.57 Ingkarni Wardli :: Farah Mohd Isa :: University of Adelaide

There have been numerous reports that the rainfall in south Western Australia, particularly around Perth has observed a step change decrease, which is typically attributed to climate change. Four statistical tests are used to assess the empirical evidence for this claim on time series from five meteorological stations, all of which exceed 50 years. The tests used in this study are: the CUSUM; Bayesian Change Point analysis; consecutive t-test and the Hotelling’s T²-statistic. Results from multivariate Hotelling’s T² analysis are compared with those from the three univariate analyses. The issue of multiple comparisons is discussed. A summary of the empirical evidence for the claimed step change in Perth area is given.
Optimal experimental design for stochastic population models
15:00 Wed 1 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Dan Pagendam :: CSIRO, Brisbane

Markov population processes are popular models for studying a wide range of phenomena including the spread of disease, the evolution of chemical reactions and the movements of organisms in population networks (metapopulations). Our ability to use these models effectively can be limited by our knowledge about parameters, such as disease transmission and recovery rates in an epidemic. Recently, there has been interest in devising optimal experimental designs for stochastic models, so that practitioners can collect data in a manner that maximises the precision of maximum likelihood estimates of the parameters for these models. I will discuss some recent work on optimal design for a variety of population models, beginning with some simple one-parameter models where the optimal design can be obtained analytically and moving on to more complicated multi-parameter models in epidemiology that involve latent states and non-exponentially distributed infectious periods. For these more complex models, the optimal design must be arrived at using computational methods and we rely on a Gaussian diffusion approximation to obtain analytical expressions for Fisher's information matrix, which is at the heart of most optimality criteria in experimental design. I will outline a simple cross-entropy algorithm that can be used for obtaining optimal designs for these models. We will also explore the improvements in experimental efficiency when using the optimal design over some simpler designs, such as the design where observations are spaced equidistantly in time.
Natural operations on the Hochschild cochain complex
13:10 Fri 3 Jun, 2011 :: Mawson 208 :: Dr Michael Batanin :: Macquarie University

The Hochschild cochain complex of an associative algebra provides an important bridge between algebra and geometry. Algebraically, this is the derived center of the algebra. Geometrically, the Hochschild cohomology of the algebra of smooth functions on a manifold is isomorphic to the graduate space of polyvector fields on this manifold. There are many important operations acting on the Hochschild complex. It is, however, a tricky question to ask which operations are natural because the Hochschild complex is not a functor. In my talk I will explain how we can overcome this obstacle and compute all possible natural operations on the Hochschild complex. The result leads immediately to a proof of the Deligne conjecture on Hochschild cochains.
Quantitative proteomics: data analysis and statistical challenges
10:10 Thu 30 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Peter Hoffmann :: Adelaide Proteomics Centre

Introduction to functional data analysis with applications to proteomics data
11:10 Thu 30 Jun, 2011 :: 7.15 Ingkarni Wardli :: A/Prof Inge Koch :: School of Mathematical Sciences

Object oriented data analysis
14:10 Thu 30 Jun, 2011 :: 7.15 Ingkarni Wardli :: Prof Steve Marron :: The University of North Carolina at Chapel Hill

Object Oriented Data Analysis is the statistical analysis of populations of complex objects. In the special case of Functional Data Analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non-Euclidean spaces, such as Lie Groups and Symmetric Spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for Object Oriented Data Analysis create several potentially large new interfaces between mathematics and statistics. Even in situations where Euclidean analysis makes sense, there are statistical challenges because of the High Dimension Low Sample Size problem, which motivates a new type of asymptotics leading to non-standard mathematical statistics.
Object oriented data analysis of tree-structured data objects
15:10 Fri 1 Jul, 2011 :: 7.15 Ingkarni Wardli :: Prof Steve Marron :: The University of North Carolina at Chapel Hill

The field of Object Oriented Data Analysis has made a lot of progress on the statistical analysis of the variation in populations of complex objects. A particularly challenging example of this type is populations of tree-structured objects. Deep challenges arise, which involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. These challenges, together with three completely different approaches to addressing them, are illustrated using a real data example, where each data point is the tree of blood arteries in one person's brain.
The (dual) local cyclic homology valued Chern-Connes character for some infinite dimensional spaces
13:10 Fri 29 Jul, 2011 :: B.19 Ingkarni Wardli :: Dr Snigdhayan Mahanta :: School of Mathematical Sciences

I will explain how to construct a bivariant Chern-Connes character on the category of sigma-C*-algebras taking values in Puschnigg's local cyclic homology. Roughly, setting the first (resp. the second) variable to complex numbers one obtains the K-theoretic (resp. dual K-homological) Chern-Connes character in one variable. We shall focus on the dual K-homological Chern-Connes character and investigate it in the example of SU(infty).
Dealing with the GC-content bias in second-generation DNA sequence data
15:10 Fri 12 Aug, 2011 :: Horace Lamb :: Prof Terry Speed :: Walter and Eliza Hall Institute

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The field of genomics is currently dealing with an explosion of data from so-called second-generation DNA sequencing machines. This is creating many challenges and opportunities for statisticians interested in the area. In this talk I will outline the technology and the data flood, and move on to one particular problem where the technology is used: copy-number analysis. There we find a novel bias, which, if not dealt with properly, can dominate the signal of interest. I will describe how we think about and summarize it, and go on to identify a plausible source of this bias, leading up to a way of removing it. Our approach makes use of the total variation metric on discrete measures, but apart from this, is largely descriptive.
Comparing Einstein to Newton via the post-Newtonian expansions
15:10 Fri 19 Aug, 2011 :: 7.15 Ingkarni Wardli :: Dr Todd Oliynyk :: Monash University

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Einstein's general relativity is presently the most accurate theory of gravity. To completely determine the gravitational field, the Einstein field equations must be solved. These equations are extremely complex and outside of a small set of idealized situations, they are impossible to solve directly. However, to make physical predictions or understand physical phenomena, it is often enough to find approximate solutions that are governed by a simpler set of equations. For example, Newtonian gravity approximates general relativity very well in regimes where the typical velocity of the gravitating matter is small compared to the speed of light. Indeed, Newtonian gravity successfully explains much of the behaviour of our solar system and is a simpler theory of gravity. However, for many situations of interest ranging from binary star systems to GPS satellites, the Newtonian approximation is not accurate enough; general relativistic effects must be included. This desire to include relativistic corrections to Newtonian gravity lead to the development of the post-Newtonian expansions.
Deformations of Oka manifolds
13:10 Fri 26 Aug, 2011 :: B.19 Ingkarni Wardli :: A/Prof Finnur Larusson :: University of Adelaide

We discuss the behaviour of the Oka property with respect to deformations of compact complex manifolds. We have recently proved that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G_delta subset of the base. We have also found a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. The special case when the fibres are tori will be considered, as well as the general case of holomorphic submersions with noncompact fibres.
Laplace's equation on multiply-connected domains
12:10 Mon 29 Aug, 2011 :: 5.57 Ingkarni Wardli :: Mr Hayden Tronnolone :: University of Adelaide

Various physical processes take place on multiply-connected domains (domains with some number of 'holes'), such as the stirring of a fluid with paddles or the extrusion of material from a die. These systems may be described by partial differential equations (PDEs). However, standard numerical methods for solving PDEs are not well-suited to such examples: finite difference methods are difficult to implement on multiply-connected domains, especially when the boundaries are irregular or moving, while finite element methods are computationally expensive. In this talk I will describe a fast and accurate numerical method for solving certain PDEs on two-dimensional multiply-connected domains, considering Laplace's equation as an example. This method takes advantage of complex variable techniques which allow the solution to be found with spectral accuracy provided the boundary data is smooth. Other advantages over traditional numerical methods will also be discussed.
Alignment of time course gene expression data sets using Hidden Markov Models
12:10 Mon 5 Sep, 2011 :: 5.57 Ingkarni Wardli :: Mr Sean Robinson :: University of Adelaide

Time course microarray experiments allow for insight into biological processes by measuring gene expression over a time period of interest. This project is concerned with time course data from a microarray experiment conducted on a particular variety of grapevine over the development of the grape berries at a number of different vineyards in South Australia. The aim of the project is to construct a methodology for combining the data from the different vineyards in order to obtain more precise estimates of the underlying behaviour of the genes over the development process. A major issue in doing so is that the rate of development of the grape berries is different at different vineyards. Hidden Markov models (HMMs) are a well established methodology for modelling time series data in a number of domains and have been previously used for gene expression analysis. Modelling the grapevine data presents a unique modelling issue, namely the alignment of the expression profiles needed to combine the data from different vineyards. In this seminar, I will describe our problem, review HMMs, present an extension to HMMs and show some preliminary results modelling the grapevine data.
Statistical analysis of metagenomic data from the microbial community involved in industrial bioleaching
12:10 Mon 19 Sep, 2011 :: 5.57 Ingkarni Wardli :: Ms Susana Soto-Rojo :: University of Adelaide

In the last two decades heap bioleaching has become established as a successful commercial option for recovering copper from low-grade secondary sulfide ores. Genetics-based approaches have recently been employed in the task of characterizing mineral processing bacteria. Data analysis is a key issue and thus the implementation of adequate mathematical and statistical tools is of fundamental importance to draw reliable conclusions. In this talk I will give a recount of two specific problems that we have been working on. The first regarding experimental design and the latter on modeling composition and activity of the microbial consortium.
Understanding the dynamics of event networks
15:00 Wed 28 Sep, 2011 :: B.18 Ingkarni Wardli :: Dr Amber Tomas :: The University of Oxford

Within many populations there are frequent communications between pairs of individuals. Such communications might be emails sent within a company, radio communications in a disaster zone or diplomatic communications between states. Often it is of interest to understand the factors that drive the observed patterns of such communications, or to study how these factors are changing over over time. Communications can be thought of as events occuring on the edges of a network which connects individuals in the population. In this talk I'll present a model for such communications which uses ideas from social network theory to account for the complex correlation structure between events. Applications to the Enron email corpus and the dynamics of hospital ward transfer patterns will be discussed.
Statistical analysis of school-based student performance data
12:10 Mon 10 Oct, 2011 :: 5.57 Ingkarni Wardli :: Ms Jessica Tan :: University of Adelaide

Join me in the journey of being a statistician for 15 minutes of your day (if you are not already one) and experience the task of data cleaning without having to get your own hands dirty. Most of you may have sat the Basic Skills Tests when at school or know someone who currently has to do the NAPLAN (National Assessment Program - Literacy and Numeracy) tests. Tests like these assess student progress and can be used to accurately measure school performance. In trying to answer the research question: "what conclusions about student progress and school performance can be drawn from NAPLAN data or data of a similar nature, using mathematical and statistical modelling and analysis techniques?", I have uncovered some interesting results about the data in my initial data analysis which I shall explain in this talk.
Statistical modelling for some problems in bioinformatics
11:10 Fri 14 Oct, 2011 :: B.17 Ingkarni Wardli :: Professor Geoff McLachlan :: The University of Queensland

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In this talk we consider some statistical analyses of data arising in bioinformatics. The problems include the detection of differential expression in microarray gene-expression data, the clustering of time-course gene-expression data and, lastly, the analysis of modern-day cytometric data. Extensions are considered to the procedures proposed for these three problems in McLachlan et al. (Bioinformatics, 2006), Ng et al. (Bioinformatics, 2006), and Pyne et al. (PNAS, 2009), respectively. The latter references are available at http://www.maths.uq.edu.au/~gjm/.
On the role of mixture distributions in the modelling of heterogeneous data
15:10 Fri 14 Oct, 2011 :: 7.15 Ingkarni Wardli :: Prof Geoff McLachlan :: University of Queensland

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We consider the role that finite mixture distributions have played in the modelling of heterogeneous data, in particular for clustering continuous data via mixtures of normal distributions. A very brief history is given starting with the seminal papers by Day and Wolfe in the sixties before the appearance of the EM algorithm. It was the publication in 1977 of the latter algorithm by Dempster, Laird, and Rubin that greatly stimulated interest in the use of finite mixture distributions to model heterogeneous data. This is because the fitting of mixture models by maximum likelihood is a classic example of a problem that is simplified considerably by the EM's conceptual unification of maximum likelihood estimation from data that can be viewed as being incomplete. In recent times there has been a proliferation of applications in which the number of experimental units n is comparatively small but the underlying dimension p is extremely large as, for example, in microarray-based genomics and other high-throughput experimental approaches. Hence there has been increasing attention given not only in bioinformatics and machine learning, but also in mainstream statistics, to the analysis of complex data in this situation where n is small relative to p. The latter part of the talk shall focus on the modelling of such high-dimensional data using mixture distributions.
Likelihood-free Bayesian inference: modelling drug resistance in Mycobacterium tuberculosis
15:10 Fri 21 Oct, 2011 :: 7.15 Ingkarni Wardli :: Dr Scott Sisson :: University of New South Wales

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A central pillar of Bayesian statistical inference is Monte Carlo integration, which is based on obtaining random samples from the posterior distribution. There are a number of standard ways to obtain these samples, provided that the likelihood function can be numerically evaluated. In the last 10 years, there has been a substantial push to develop methods that permit Bayesian inference in the presence of computationally intractable likelihood functions. These methods, termed ``likelihood-free'' or approximate Bayesian computation (ABC), are now being applied extensively across many disciplines. In this talk, I'll present a brief, non-technical overview of the ideas behind likelihood-free methods. I'll motivate and illustrate these ideas through an analysis of the epidemiological fitness cost of drug resistance in Mycobacterium tuberculosis.
Dirac operators on classifying spaces
13:10 Fri 28 Oct, 2011 :: B.19 Ingkarni Wardli :: Dr Pedram Hekmati :: University of Adelaide

The Dirac operator was introduced by Paul Dirac in 1928 as the formal square root of the D'Alembert operator. Thirty years later it was rediscovered in Euclidean signature by Atiyah and Singer in their seminal work on index theory. In this talk I will describe efforts to construct a Dirac type operator on the classifying space for odd complex K-theory. Ultimately the aim is to produce a projective family of Fredholm operators realising elements in twisted K-theory of a certain moduli stack.
Mathematical opportunities in molecular space
15:10 Fri 28 Oct, 2011 :: B.18 Ingkarni Wardli :: Dr Aaron Thornton :: CSIRO

The study of molecular motion, interaction and space at the nanoscale has become a powerful tool in the area of gas separation, storage and conversion for efficient energy solutions. Modeling in this field has typically involved highly iterative computational algorithms such as molecular dynamics, Monte Carlo and quantum mechanics. Mathematical formulae in the form of analytical solutions to this field offer a range of useful and insightful advantages including optimization, bifurcation analysis and standardization. Here we present a few case scenarios where mathematics has provided insight and opportunities for further investigation.
Metric geometry in data analysis
13:10 Fri 11 Nov, 2011 :: B.19 Ingkarni Wardli :: Dr Facundo Memoli :: University of Adelaide

The problem of object matching under invariances can be studied using certain tools from metric geometry. The central idea is to regard objects as metric spaces (or metric measure spaces). The type of invariance that one wishes to have in the matching is encoded by the choice of the metrics with which one endows the objects. The standard example is matching objects in Euclidean space under rigid isometries: in this situation one would endow the objects with the Euclidean metric. More general scenarios are possible in which the desired invariance cannot be reflected by the preservation of an ambient space metric. Several ideas due to M. Gromov are useful for approaching this problem. The Gromov-Hausdorff distance is a natural candidate for doing this. However, this metric leads to very hard combinatorial optimization problems and it is difficult to relate to previously reported practical approaches to the problem of object matching. I will discuss different variations of these ideas, and in particular will show a construction of an L^p version of the Gromov-Hausdorff metric, called the Gromov-Wassestein distance, which is based on mass transportation ideas. This new metric directly leads to quadratic optimization problems on continuous variables with linear constraints. As a consequence of establishing several lower bounds, it turns out that several invariants of metric measure spaces turn out to be quantitatively stable in the GW sense. These invariants provide practical tools for the discrimination of shapes and connect the GW ideas to a number of pre-existing approaches.
Stability analysis of nonparallel unsteady flows via separation of variables
15:30 Fri 18 Nov, 2011 :: 7.15 Ingkarni Wardli :: Prof Georgy Burde :: Ben-Gurion University

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The problem of variables separation in the linear stability equations, which govern the disturbance behavior in viscous incompressible fluid flows, is discussed. Stability of some unsteady nonparallel three-dimensional flows (exact solutions of the Navier-Stokes equations) is studied via separation of variables using a semi-analytical, semi-numerical approach. In this approach, a solution with separated variables is defined in a new coordinate system which is sought together with the solution form. As the result, the linear stability problems are reduced to eigenvalue problems for ordinary differential equations which can be solved numerically. In some specific cases, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for methods used in the hydrodynamic stability theory. Exact solutions of the stability problems for some stagnation-type flows are presented.
Fluid flows in microstructured optical fibre fabrication
15:10 Fri 25 Nov, 2011 :: B.17 Ingkarni Wardli :: Mr Hayden Tronnolone :: University of Adelaide

Optical fibres are used extensively in modern telecommunications as they allow the transmission of information at high speeds. Microstructured optical fibres are a relatively new fibre design in which a waveguide for light is created by a series of air channels running along the length of the material. The flexibility of this design allows optical fibres to be created with adaptable (and previously unrealised) optical properties. However, the fluid flows that arise during fabrication can greatly distort the geometry, which can reduce the effectiveness of a fibre or render it useless. I will present an overview of the manufacturing process and highlight the difficulties. I will then focus on surface-tension driven deformation of the macroscopic version of the fibre extruded from a reservoir of molten glass, occurring during fabrication, which will be treated as a two-dimensional Stokes flow problem. I will outline two different complex-variable numerical techniques for solving this problem along with comparisons of the results, both to other models and to experimental data.
Collision and instability in a rotating fluid-filled torus
15:10 Mon 12 Dec, 2011 :: Benham Lecture Theatre :: Dr Richard Clarke :: The University of Auckland

The simple experiment discussed in this talk, first conceived by Madden and Mullin (JFM, 1994) as part of their investigations into the non-uniqueness of decaying turbulent flow, consists of a fluid-filled torus which is rotated in an horizontal plane. Turbulence within the contained flow is triggered through a rapid change in its rotation rate. The flow instabilities which transition the flow to this turbulent state, however, are truly fascinating in their own right, and form the subject of this presentation. Flow features observed in both UK- and Auckland-based experiments will be highlighted, and explained through both boundary-layer analysis and full DNS. In concluding we argue that this flow regime, with its compact geometry and lack of cumbersome flow entry effects, presents an ideal regime in which to study many prototype flow behaviours, very much in the same spirit as Taylor-Couette flow.
Plurisubharmonic subextensions as envelopes of disc functionals
13:10 Fri 2 Mar, 2012 :: B.20 Ingkarni Wardli :: A/Prof Finnur Larusson :: University of Adelaide

I will describe new joint work with Evgeny Poletsky. We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient is a complex manifold with a local biholomorphism to $X$, except it need not be Hausdorff. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.
The de Rham Complex
12:10 Mon 19 Mar, 2012 :: 5.57 Ingkarni Wardli :: Mr Michael Albanese :: University of Adelaide

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The de Rham complex is of fundamental importance in differential geometry. After first introducing differential forms (in the familiar setting of Euclidean space), I will demonstrate how the de Rham complex elegantly encodes one half (in a sense which will become apparent) of the results from vector calculus. If there is time, I will indicate how results from the remaining half of the theory can be concisely expressed by a single, far more general theorem.
Financial risk measures - the theory and applications of backward stochastic difference/differential equations with respect to the single jump process
12:10 Mon 26 Mar, 2012 :: 5.57 Ingkarni Wardli :: Mr Bin Shen :: University of Adelaide

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This is my PhD thesis submitted one month ago. Chapter 1 introduces the backgrounds of the research fields. Then each chapter is a published or an accepted paper. Chapter 2, to appear in Methodology and Computing in Applied Probability, establishes the theory of Backward Stochastic Difference Equations with respect to the single jump process in discrete time. Chapter 3, published in Stochastic Analysis and Applications, establishes the theory of Backward Stochastic Differential Equations with respect to the single jump process in continuous time. Chapter 2 and 3 consist of Part I Theory. Chapter 4, published in Expert Systems With Applications, gives some examples about how to measure financial risks by the theory established in Chapter 2. Chapter 5, accepted by Journal of Applied Probability, considers the question of an optimal transaction between two investors to minimize their risks. It's the applications of the theory established in Chapter 3. Chapter 4 and 5 consist of Part II Applications.
Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds
13:10 Fri 4 May, 2012 :: Napier LG28 :: Dr Tyson Ritter :: University of Adelaide

In complex geometry a manifold is Stein if there are, in a certain sense, "many" holomorphic maps from the manifold into C^n. While this has long been well understood, a fruitful definition of the dual notion has until recently been elusive. In Oka theory, a manifold is Oka if it satisfies several equivalent definitions, each stating that the manifold has "many" holomorphic maps into it from C^n. Related to this is the geometric condition of ellipticity due to Gromov, who showed that it implies a complex manifold is Oka. We present recent contributions to three open questions involving elliptic and Oka manifolds. We show that affine quotients of C^n are elliptic, and combine this with an example of Margulis to construct new elliptic manifolds of interesting homotopy types. It follows that every open Riemann surface properly acyclically embeds into an elliptic manifold, extending an existing result for open Riemann surfaces with abelian fundamental group.
Are Immigrants Discriminated in the Australian Labour Market?
12:10 Mon 7 May, 2012 :: 5.57 Ingkarni Wardli :: Ms Wei Xian Lim :: University of Adelaide

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In this talk, I will present what I did in my honours project, which was to determine if immigrants, categorised as immigrants from English speaking countries and Non-English speaking countries, are discriminated in the Australian labour market. To determine if discrimination exists, a decomposition of the wage function is applied and analysed via regression analysis. Two different methods of estimating the unknown parameters in the wage function will be discussed: 1. the Ordinary Least Square method, 2. the Quantile Regression method. This is your rare chance of hearing me talk about non-nanomathematics related stuff!
Change detection in rainfall times series for Perth, Western Australia
12:10 Mon 14 May, 2012 :: 5.57 Ingkarni Wardli :: Ms Farah Mohd Isa :: University of Adelaide

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There have been numerous reports that the rainfall in south Western Australia, particularly around Perth has observed a step change decrease, which is typically attributed to climate change. Four statistical tests are used to assess the empirical evidence for this claim on time series from five meteorological stations, all of which exceed 50 years. The tests used in this study are: the CUSUM; Bayesian Change Point analysis; consecutive t-test and the Hotelling's T^2-statistic. Results from multivariate Hotelling's T^2 analysis are compared with those from the three univariate analyses. The issue of multiple comparisons is discussed. A summary of the empirical evidence for the claimed step change in Perth area is given.
Geometric modular representation theory
13:10 Fri 1 Jun, 2012 :: Napier LG28 :: Dr Anthony Henderson :: University of Sydney

Representation theory is one of the oldest areas of algebra, but many basic questions in it are still unanswered. This is especially true in the modular case, where one considers vector spaces over a field F of positive characteristic; typically, complications arise for particular small values of the characteristic. For example, from a vector space V one can construct the symmetric square S^2(V), which is one easy example of a representation of the group GL(V). One would like to say that this representation is irreducible, but that statement is not always true: if F has characteristic 2, there is a nontrivial invariant subspace. Even for GL(V), we do not know the dimensions of all irreducible representations in all characteristics. In this talk, I will introduce some of the main ideas of geometric modular representation theory, a more recent approach which is making progress on some of these old problems. Essentially, the strategy is to re-formulate everything in terms of homology of various topological spaces, where F appears only as the field of coefficients and the spaces themselves are independent of F; thus, the modular anomalies in representation theory arise because homology with modular coefficients is detecting something about the topology that rational coefficients do not. In practice, the spaces are usually varieties over the complex numbers, and homology is replaced by intersection cohomology to take into account the singularities of these varieties.
Enhancing the Jordan canonical form
15:10 Fri 1 Jun, 2012 :: B.21 Ingkarni Wardli :: A/Prof Anthony Henderson :: The University of Sydney

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In undergraduate linear algebra, we teach the Jordan canonical form theorem: that every similarity class of n x n complex matrices contains a special matrix which is block-diagonal with each block having a very simple form (a single eigenvalue repeated down the diagonal, ones on the super-diagonal, and zeroes elsewhere). This is of course very useful for matrix calculations. After explaining some of the general context of this result, I will focus on a case which, despite its close proximity to the Jordan canonical form theorem, has only recently been worked out: the classification of pairs of a vector and a matrix.
Model turbulent floods based upon the Smagorinski large eddy closure
12:10 Mon 4 Jun, 2012 :: 5.57 Ingkarni Wardli :: Mr Meng Cao :: University of Adelaide

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Rivers, floods and tsunamis are often very turbulent. Conventional models of such environmental fluids are typically based on depth-averaged inviscid irrotational flow equations. We explore changing such a base to the turbulent Smagorinski large eddy closure. The aim is to more appropriately model the fluid dynamics of such complex environmental fluids by using such a turbulent closure. Large changes in fluid depth are allowed. Computer algebra constructs the slow manifold of the flow in terms of the fluid depth h and the mean turbulent lateral velocities u and v. The major challenge is to deal with the nonlinear stress tensor in the Smagorinski closure. The model integrates the effects of inertia, self-advection, bed drag, gravitational forcing and turbulent dissipation with minimal assumptions. Although the resultant model is close to established models, the real outcome is creating a sound basis for the modelling so others, in their modelling of more complex situations, can systematically include more complex physical processes.
Introduction to quantales via axiomatic analysis
13:10 Fri 15 Jun, 2012 :: Napier LG28 :: Dr Ittay Weiss :: University of the South Pacific

Quantales were introduced by Mulvey in 1986 in the context of non-commutative topology with the aim of providing a concrete non-commutative framework for the foundations of quantum mechanics. Since then quantales found applications in other areas as well, among others in the work of Flagg. Flagg considers certain special quantales, called value quantales, that are desigend to capture the essential properties of ([0,\infty],\le,+) that are relevant for analysis. The result is a well behaved theory of value quantale enriched metric spaces. I will introduce the notion of quantales as if they were desigend for just this purpose, review most of the known results (since there are not too many), and address a some new results, conjectures, and questions.
Complex geometry and operator theory
14:10 Mon 9 Jul, 2012 :: Ingkarni Wardli B19 :: Prof Ron Douglas :: Texas A&M University

In the study of bounded operators on Hilbert spaces of holomorphic functions, concepts and techniques from complex geometry are important. An anti-holomorphic bundle exists on which one can define the Chern connection. Its curvature turns out to be a complete invariant and various operator notions can't be reframed in terms of geometrical ones which leads to the solution of some problems. We will discuss this approach with an emphasis on natural examples in the one and multivariable case.
The motivic logarithm and its realisations
13:10 Fri 3 Aug, 2012 :: Engineering North 218 :: Dr James Borger :: Australian National University

When a complex manifold is defined by polynomial equations, its cohomology groups inherit extra structure. This was discovered by Hodge in the 1920s and 30s. When the defining polynomials have rational coefficients, there is some additional, arithmetic structure on the cohomology. This was discovered by Grothendieck and others in the 1960s. But here the situation is still quite mysterious because each cohomology group has infinitely many different arithmetic structures and while they are not directly comparable, they share many properties---with each other and with the Hodge structure. All written accounts of this that I'm aware of treat arbitrary varieties. They are beautifully abstract and non-explicit. In this talk, I'll take the opposite approach and try to give a flavour of the subject by working out a perhaps the simplest nontrivial example, the cohomology of C* relative to a subset of two points, in beautifully concrete and explicit detail. Here the common motif is the logarithm. In Hodge theory, it is realised as the complex logarithm; in the crystalline theory, it's as the p-adic logarithm; and in the etale theory, it's as Kummer theory. I'll assume you have some familiarity with usual, singular cohomology of topological spaces, but I won't assume that you know anything about these non-topological cohomology theories.
Geometry - algebraic to arithmetic to absolute
15:10 Fri 3 Aug, 2012 :: B.21 Ingkarni Wardli :: Dr James Borger :: Australian National University

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Classical algebraic geometry is about studying solutions to systems of polynomial equations with complex coefficients. In arithmetic algebraic geometry, one digs deeper and studies the arithmetic properties of the solutions when the coefficients are rational, or even integral. From the usual point of view, it's impossible to go deeper than this for the simple reason that no smaller rings are available - the integers have no proper subrings. In this talk, I will explain how an emerging subject, lambda-algebraic geometry, allows one to do just this and why one might care.
Hodge numbers and cohomology of complex algebraic varieties
13:10 Fri 10 Aug, 2012 :: Engineering North 218 :: Prof Gus Lehrer :: University of Sydney

Let $X$ be a complex algebraic variety defined over the ring $\mathfrak{O}$ of integers in a number field $K$ and let $\Gamma$ be a group of $\mathfrak{O}$-automorphisms of $X$. I shall discuss how the counting of rational points over reductions mod $p$ of $X$, and an analysis of the Hodge structure of the cohomology of $X$, may be used to determine the cohomology as a $\Gamma$-module. This will include some joint work with Alex Dimca and with Mark Kisin, and some classical unsolved problems.
Drawing of Viscous Threads with Temperature-dependent Viscosity
14:10 Fri 10 Aug, 2012 :: Engineering North N218 :: Dr Jonathan Wylie :: City University of Hong Kong

The drawing of viscous threads is important in a wide range of industrial applications and is a primary manufacturing process in the optical fiber and textile industries. Most of the materials used in these processes have viscosities that vary extremely strongly with temperature. We investigate the role played by viscous heating in the drawing of viscous threads. Usually, the effects of viscous heating and inertia are neglected because the parameters that characterize them are typically very small. However, by performing a detailed theoretical analysis we surprisingly show that even very small amounts of viscous heating can lead to a runaway phenomena. On the other hand, inertia prevents runaway, and the interplay between viscous heating and inertia results in very complicated dynamics for the system. Even more surprisingly, in the absence of viscous heating, we find that a new type of instability can occur when a thread is heated by a radiative heat source. By analyzing an asymptotic limit of the Navier-Stokes equation we provide a theory that describes the nature of this instability and explains the seemingly counterintuitive behavior.
Air-cooled binary Rankine cycle performance with varying ambient temperature
12:10 Mon 13 Aug, 2012 :: B.21 Ingkarni Wardli :: Ms Josephine Varney :: University of Adelaide

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Next month, I have to give a presentation in Reno, Nevada to a group of geologists, engineers and geophysicists. So, for this talk, I am going to ask you to pretend you know very little about maths (and perhaps a lot about geology) and give me some feedback on my proposed talk. The presentation itself, is about the effect of air-cooling on geothermal power plant performance. Air-cooling is necessary for geothermal plays in dry areas, and ambient air temperature significantly affects the power output of air-cooled geothermal power plants. Hence, a method for determining the effect of ambient air temperature on geothermal power plants is presented. Using the ambient air temperature distribution from Leigh Creek, South Australia, this analysis shows that an optimally designed plant produces 6% more energy annually than a plant designed using the mean ambient temperature.
Differential topology 101
13:10 Fri 17 Aug, 2012 :: Engineering North 218 :: Dr Nicholas Buchdahl :: University of Adelaide

Much of my recent research been directed at a problem in the theory of compact complex surfaces---trying to fill in a gap in the Enriques-Kodaira classification. Attempting to classify some collection of mathematical objects is a very common activity for pure mathematicians, and there are many well-known examples of successful classification schemes; for example, the classification of finite simple groups, and the classification of simply connected topological 4-manifolds. The aim of this talk will be to illustrate how techniques from differential geometry can be used to classify compact surfaces. The level of the talk will be very elementary, and the material is all very well known, but it is sometimes instructive to look back over simple cases of a general problem with the benefit of experience to gain greater insight into the more general and difficult cases.
Star Wars Vs The Lord of the Rings: A Survival Analysis
12:10 Mon 27 Aug, 2012 :: B.21 Ingkarni Wardli :: Mr Christopher Davies :: University of Adelaide

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Ever wondered whether you are more likely to die in the Galactic Empire or Middle Earth? Well this is the postgraduate seminar for you! I'll be attempting to answer this question using survival analysis, the statistical method of choice for investigating time to event data. Spoiler Warning: This talk will contain references to the deaths of characters in the above movie sagas.
Holomorphic flexibility properties of compact complex surfaces
13:10 Fri 31 Aug, 2012 :: Engineering North 218 :: A/Prof Finnur Larusson :: University of Adelaide

I will describe recent joint work with Franc Forstneric (arXiv, July 2012). We introduce a new property, called the stratified Oka property, which fits into a hierarchy of anti-hyperbolicity properties that includes the Oka property. We show that stratified Oka manifolds are strongly dominable by affine spaces. It follows that Kummer surfaces are strongly dominable. We determine which minimal surfaces of class VII are Oka (assuming the global spherical shell conjecture). We deduce that the Oka property and several other anti-hyperbolicity properties are in general not closed in families of compact complex manifolds. I will summarise what is known about how the Oka property fits into the Enriques-Kodaira classification of surfaces.
Principal Component Analysis (PCA)
12:30 Mon 3 Sep, 2012 :: B.21 Ingkarni Wardli :: Mr Lyron Winderbaum :: University of Adelaide

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Principal Component Analysis (PCA) has become something of a buzzword recently in a number of disciplines including the gene expression and facial recognition. It is a classical, and fundamentally simple, concept that has been around since the early 1900's, its recent popularity largely due to the need for dimension reduction techniques in analyzing high dimensional data that has become more common in the last decade, and the availability of computing power to implement this. I will explain the concept, prove a result, and give a couple of examples. The talk should be accessible to all disciplines as it (should?) only assume first year linear algebra, the concept of a random variable, and covariance.
Electrokinetics of concentrated suspensions of spherical particles
15:10 Fri 28 Sep, 2012 :: B.21 Ingkarni Wardli :: Dr Bronwyn Bradshaw-Hajek :: University of South Australia

Electrokinetic techniques are used to gather specific information about concentrated dispersions such as electronic inks, mineral processing slurries, pharmaceutical products and biological fluids (e.g. blood). But, like most experimental techniques, intermediate quantities are measured, and consequently the method relies explicitly on theoretical modelling to extract the quantities of experimental interest. A self-consistent cell-model theory of electrokinetics can be used to determine the electrical conductivity of a dense suspension of spherical colloidal particles, and thereby determine the quantities of interest (such as the particle surface potential). The numerical predictions of this model compare well with published experimental results. High frequency asymptotic analysis of the cell-model leads to some interesting conclusions.
Turbulent flows, semtex, and rainbows
12:10 Mon 8 Oct, 2012 :: B.21 Ingkarni Wardli :: Ms Sophie Calabretto :: University of Adelaide

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The analysis of turbulence in transient flows has applications across a broad range of fields. We use the flow of fluid in a toroidal container as a paradigm for studying the complex dynamics due to this turbulence. To explore the dynamics of our system, we exploit the numerical capabilities of semtex; a quadrilateral spectral element DNS code. Rainbows result.
Complex analysis in low Reynolds number hydrodynamics
15:10 Fri 12 Oct, 2012 :: B.20 Ingkarni Wardli :: Prof Darren Crowdy :: Imperial College London

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It is a well-known fact that the methods of complex analysis provide great advantage in studying physical problems involving a harmonic field satisfying Laplace's equation. One example is in ideal fluid mechanics (infinite Reynolds number) where the absence of viscosity, and the assumption of zero vorticity, mean that it is possible to introduce a so-called complex potential -- an analytic function from which all physical quantities of interest can be inferred. In the opposite limit of zero Reynolds number flows which are slow and viscous and the governing fields are not harmonic it is much less common to employ the methods of complex analysis even though they continue to be relevant in certain circumstances. This talk will give an overview of a variety of problems involving slow viscous Stokes flows where complex analysis can be usefully employed to gain theoretical insights. A number of example problems will be considered including the locomotion of low-Reynolds-number micro-organisms and micro-robots, the friction properties of superhydrophobic surfaces in microfluidics and problems of viscous sintering and the manufacture of microstructured optic fibres (MOFs).
AD Model Builder and the estimation of lobster abundance
12:10 Mon 22 Oct, 2012 :: B.21 Ingkarni Wardli :: Mr John Feenstra :: University of Adelaide

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Determining how many millions of lobsters reside in our waters and how it changes over time is a central aim of lobster stock assessment. ADMB is powerful optimisation software to model and solve complex non-linear problems using automatic differentiation and plays a major role in SA and worldwide in fisheries stock assessment analyses. In this talk I will provide a brief description of an example modelling problem, key features and use of ADMB.
The space of cubic rational maps
13:10 Fri 26 Oct, 2012 :: Engineering North 218 :: Mr Alexander Hanysz :: University of Adelaide

For each natural number d, the space of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the degree 3 case, studying a double action of the Mobius group on the space of cubic rational maps. We show that the categorical quotient is C, and that the space of cubic rational maps enjoys the holomorphic flexibility properties of strong dominability and C-connectedness.
Numerical Free Probability: Computing Eigenvalue Distributions of Algebraic Manipulations of Random Matrices
15:10 Fri 2 Nov, 2012 :: B.20 Ingkarni Wardli :: Dr Sheehan Olver :: The University of Sydney

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Suppose that the global eigenvalue distributions of two large random matrices A and B are known. It is a remarkable fact that, generically, the eigenvalue distribution of A + B and (if A and B are positive definite) A*B are uniquely determined from only the eigenvalue distributions of A and B; i.e., no information about eigenvectors are required. These operations on eigenvalue distributions are described by free probability theory. We construct a numerical toolbox that can efficiently and reliably calculate these operations with spectral accuracy, by exploiting the complex analytical framework that underlies free probability theory.
Variation of Hodge structure for generalized complex manifolds
13:10 Fri 7 Dec, 2012 :: Ingkarni Wardli B20 :: Dr David Baraglia :: University of Adelaide

Generalized complex geometry combines complex and symplectic geometry into a single framework, incorporating also holomorphic Poisson and bi-Hermitian structures. The Dolbeault complex naturally extends to the generalized complex setting giving rise to Hodge structures in twisted cohomology. We consider the variations of Hodge structure and period mappings that arise from families of generalized complex manifolds. As an application we prove a local Torelli theorem for generalized Calabi-Yau manifolds.
Recent results on holomorphic extension of functions on unbounded domains in C^n
11:10 Fri 21 Dec, 2012 :: Ingkarni Wardli B19 :: Prof Roman Dwilewicz :: Missouri University of Science and Technology

In the talk there will be given a short review of holomorphic extension problems starting with the famous Hartogs theorem (1906) up to recent results on global holomorphic extensions for unbounded domains, obtained together with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago). There is an interesting geometry behind the extension problem for unbounded domains, namely (in some cases) it depends on the position of a complex variety in the closure of the domain. The extension problem appeared non-trivial and the work is in progress. However the talk will be illustrated by many figures and pictures and should be accessible also to graduate students.
Twistor theory and the harmonic hull
15:10 Fri 8 Mar, 2013 :: B.18 Ingkarni Wardli :: Prof Michael Eastwood :: Australian National University

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Harmonic functions are real-analytic and so automatically extend as functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated conformal geometry. Nothing will be supposed about such matters: I shall base the constructions on an elementary yet mysterious formula of Bateman from 1904. This is joint work with Feng Xu.
On the chromatic number of a random hypergraph
13:10 Fri 22 Mar, 2013 :: Ingkarni Wardli B21 :: Dr Catherine Greenhill :: University of New South Wales

A hypergraph is a set of vertices and a set of hyperedges, where each hyperedge is a subset of vertices. A hypergraph is r-uniform if every hyperedge contains r vertices. A colouring of a hypergraph is an assignment of colours to vertices such that no hyperedge is monochromatic. When the colours are drawn from the set {1,..,k}, this defines a k-colouring. We consider the problem of k-colouring a random r-uniform hypergraph with n vertices and cn edges, where k, r and c are constants and n tends to infinity. In this setting, Achlioptas and Naor showed that for the case of r = 2, the chromatic number of a random graph must have one of two easily computable values as n tends to infinity. I will describe some joint work with Martin Dyer (Leeds) and Alan Frieze (Carnegie Mellon), in which we generalised this result to random uniform hypergraphs. The argument uses the second moment method, and applies a general theorem for performing Laplace summation over a lattice. So the proof contains something for everyone, with elements from combinatorics, analysis and algebra.
A stability theorem for elliptic Harnack inequalities
15:10 Fri 5 Apr, 2013 :: B.18 Ingkarni Wardli :: Prof Richard Bass :: University of Connecticut

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Harnack inequalities are an important tool in probability theory, analysis, and partial differential equations. The classical Harnack inequality is just the one you learned in your graduate complex analysis class, but there have been many extensions, to different spaces, such as manifolds, fractals, infinite graphs, and to various sorts of elliptic operators. A landmark result was that of Moser in 1961, where he proved the Harnack inequality for solutions to a class of partial differential equations. I will talk about the stability of Harnack inequalities. The main result says that if the Harnack inequality holds for an operator on a space, then the Harnack inequality will also hold for a large class of other operators on that same space. This provides a generalization of the result of Moser.
An Oka principle for equivariant isomorphisms
12:10 Fri 3 May, 2013 :: Ingkarni Wardli B19 :: A/Prof Finnur Larusson :: University of Adelaide

I will discuss new joint work with Frank Kutzschebauch (Bern) and Gerald Schwarz (Brandeis). Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$? In a situation that we describe, with some justification, as generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if there is a $G$-diffeomorphism $X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even in simply defined examples. Our work is in part motivated by the linearisation problem for actions on $\C^n$. It follows from one of our main results that a holomorphic $G$-action on $\C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.
Models of cell-extracellular matrix interactions in tissue engineering
15:10 Fri 3 May, 2013 :: B.18 Ingkarni Wardli :: Dr Ed Green :: University of Adelaide

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Tissue engineers hope in future to be able to grow functional tissues in vitro to replace those that are damaged by injury, disease, or simple wear and tear. They use cell culture methods, such as seeding cells within collagen gels, that are designed to mimic the cells' environment in vivo. Amongst other factors, it is clear that mechanical interactions between cells and the extracellular matrix (ECM) in which they reside play an important role in tissue development. However, the mechanics of the ECM is complex, and at present, its role is only partly understood. In this talk, I will present mathematical models of some simple cell-ECM interaction problems, and show how they can be used to gain more insight into the processes that regulate tissue development.
Pulsatile Flow
12:10 Mon 20 May, 2013 :: B.19 Ingkarni Wardli :: David Wilke :: University of Adelaide

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Blood flow within the human arterial system is inherently unsteady as a consequence of the pulsations of the heart. The unsteady nature of the flow gives rise to a number of important flow features which may be critical in understanding pathologies of the cardiovascular system. For example, it is believed that large oscillations in wall shear stress may enhance the effects of artherosclerosis, among other pathologies. In this talk I will present some of the basic concepts of pulsatile flow and follow the analysis first performed by J.R. Womersley in his seminal 1955 paper.
Multiscale modelling couples patches of wave-like simulations
12:10 Mon 27 May, 2013 :: B.19 Ingkarni Wardli :: Meng Cao :: University of Adelaide

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A multiscale model is proposed to significantly reduce the expensive numerical simulations of complicated waves over large spatial domains. The multiscale model is built from given microscale simulations of complicated physical processes such as sea ice or turbulent shallow water. Our long term aim is to enable macroscale simulations obtained by coupling small patches of simulations together over large physical distances. This initial work explores the coupling of patch simulations of wave-like pdes. With the line of development being to water waves we discuss the dynamics of two complementary fields called the 'depth' h and 'velocity' u. A staggered grid is used for the microscale simulation of the depth h and velocity u. We introduce a macroscale staggered grid to couple the microscale patches. Linear or quadratic interpolation provides boundary conditions on the field in each patch. Linear analysis of the whole coupled multiscale system establishes that the resultant macroscale dynamics is appropriate. Numerical simulations support the linear analysis. This multiscale method should empower the feasible computation of large scale simulations of wave-like dynamics with complicated underlying physics.
Fire-Atmosphere Models
12:10 Mon 29 Jul, 2013 :: B.19 Ingkarni Wardli :: Mika Peace :: University of Adelaide

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Fire behaviour models are increasingly being used to assist in planning and operational decisions for bush fires and fuel reduction burns. Rate of spread (ROS) of the fire front is a key output of such models. The ROS value is typically calculated from a formula which has been derived from empirical data, using very simple meteorological inputs. We have used a coupled fire-atmosphere model to simulate real bushfire events. The results show that complex interactions between a fire and the atmosphere can have a significant influence on fire spread, thus highlighting the limitations of a model that uses simple meteorological inputs.
Geometry of moduli spaces
12:10 Fri 30 Aug, 2013 :: Ingkarni Wardli B19 :: Prof Georg Schumacher :: University of Marburg

We discuss the concept of moduli spaces in complex geometry. The main examples are moduli of compact Riemann surfaces, moduli of compact projective varieties and moduli of holomorphic vector bundles, whose points correspond to isomorphism classes of the given objects. Moduli spaces carry a natural topology, whereas a complex structure that reflects the variation of the structure in a family exists in general only under extra conditions. In a similar way, a natural hermitian metric (Weil-Petersson metric) on moduli spaces that induces a symplectic structure can be constructed from the variation of distinguished metrics on the fibers. In this way, various questions concerning the underlying symplectic structure, the curvature of the Weil-Petersson metric, hyperbolicity of moduli spaces, and construction of positive/ample line bundles on compactified moduli spaces can be answered.
Medical Decision Analysis
12:10 Mon 2 Sep, 2013 :: B.19 Ingkarni Wardli :: Eka Baker :: University of Adelaide

Doctors make life changing decisions every day based on clinical trial data. However, this data is often obtained from studies on healthy individuals or on patients with only the disease that a treatment is targeting. Outside of these studies, many patients will have other conditions that may affect the predicted benefit of receiving a certain treatment. I will talk about what clinical trials are, how to measure the benefit of treatments, and how having multiple conditions (comorbidities) will affect the benefit of treatments.
Dynamics and the geometry of numbers
14:10 Fri 27 Sep, 2013 :: Horace Lamb Lecture Theatre :: Prof Akshay Venkatesh :: Stanford University

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It was understood by Minkowski that one could prove interesting results in number theory by considering the geometry of lattices in R^n. (A lattice is simply a grid of points.) This technique is called the "geometry of numbers." We now understand much more about analysis and dynamics on the space of all lattices, and this has led to a deeper understanding of classical questions. I will review some of these ideas, with emphasis on the dynamical aspects.
Gravitational slingshot and space mission design
15:10 Fri 11 Oct, 2013 :: B.18 Ingkarni Wardli :: Prof Pawel Nurowski :: Polish Academy of Sciences

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When planning a space mission the weight of the spacecraft is the main issue. Every gram sent into the outer space costs a lot. A considerable part of the overall weight of the spaceship consists of a fuel needed to control it. I will explain how space agencies reduce the amount of fuel needed to go to a given place in the Solar System by using gravity of celestial bodies encountered along the trip. I will start with the explanation of an old trick called `gravitational slingshot', and end up with a modern technique which is based on the analysis of a 3-body problem appearing in Newtonian mechanics.
Classification Using Censored Functional Data
15:10 Fri 18 Oct, 2013 :: B.18 Ingkarni Wardli :: A/Prof Aurore Delaigle :: University of Melbourne

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We consider classification of functional data. This problem has received a lot of attention in the literature in the case where the curves are all observed on the same interval. A difficulty in applications is that the functional curves can be supported on quite different intervals, in which case standard methods of analysis cannot be used. We are interested in constructing classifiers for curves of this type. More precisely, we consider classification of functions supported on a compact interval, in cases where the training sample consists of functions observed on other intervals, which may differ among the training curves. We propose several methods, depending on whether or not the observable intervals overlap by a significant amount. In the case where these intervals differ a lot, our procedure involves extending the curves outside the interval where they were observed. We suggest a new nonparametric approach for doing this. We also introduce flexible ways of combining potential differences in shapes of the curves from different populations, and potential differences between the endpoints of the intervals where the curves from each population are observed.
All at sea with spectral analysis
11:10 Tue 19 Nov, 2013 :: Ingkarni Wardli Level 5 Room 5.56 :: A/Prof Andrew Metcalfe :: The University of Adelaide

The steady state response of a single degree of freedom damped linear stystem to a sinusoidal input is a sinusoidal function at the same frequency, but generally with a different amplitude and a phase shift. The analogous result for a random stationary input can be described in terms of input and response spectra and a transfer function description of the linear system. The practical use of this result is that the parameters of a linear system can be estimated from the input and response spectra, and the response spectrum can be predicted if the transfer function and input spectrum are known. I shall demonstrate these results with data from a small ship in the North Sea. The results from the sea trial raise the issue of non-linearity, and second order amplitude response functons are obtained using auto-regressive estimators. The possibility of using wavelets rather than spectra is consedred in the context of single degree of freedom linear systems. Everybody welcome to attend. Please not a change of venue - we will be in room 5.56
Reductive group actions and some problems concerning their quotients
12:10 Fri 17 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Gerald Schwarz :: Brandeis University

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We will gently introduce the concept of a complex reductive group and the notion of the quotient Z of a complex vector space V on which our complex reductive group G acts linearly. There is the quotient mapping p from V to Z. The quotient is an affine variety with a stratification coming from the group action. Let f be an automorphism of Z. We consider the following questions (and give some answers). 1) Does f preserve the stratification of Z, i.e., does it permute the strata? 2) Is there a lift F of f? This means that F maps V to V and p(F(v))=f(p(v)) for all v in V. 3) Can we arrange that F is equivariant? We show that 1) is almost always true, that 2) is true in a lot of cases and that a twisted version of 3) then holds.
The density property for complex manifolds: a strong form of holomorphic flexibility
12:10 Fri 24 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Frank Kutzschebauch :: University of Bern

Compared with the real differentiable case, complex manifolds in general are more rigid, their groups of holomorphic diffeomorphisms are rather small (in general trivial). A long known exception to this behavior is affine n-space C^n for n at least 2. Its group of holomorphic diffeomorphisms is infinite dimensional. In the late 1980s Andersen and Lempert proved a remarkable theorem which stated in its generalized version due to Forstneric and Rosay that any local holomorphic phase flow given on a Runge subset of C^n can be locally uniformly approximated by a global holomorphic diffeomorphism. The main ingredient in the proof was formalized by Varolin and called the density property: The Lie algebra generated by complete holomorphic vector fields is dense in the Lie algebra of all holomorphic vector fields. In these manifolds a similar local to global approximation of Andersen-Lempert type holds. It is a precise way of saying that the group of holomorphic diffeomorphisms is large. In the talk we will explain how this notion is related to other more recent flexibility notions in complex geometry, in particular to the notion of a Oka-Forstneric manifold. We will give examples of manifolds with the density property and sketch applications of the density property. If time permits we will explain criteria for the density property developed by Kaliman and the speaker.
Holomorphic null curves and the conformal Calabi-Yau problem
12:10 Tue 28 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Franc Forstneric :: University of Ljubljana

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I shall describe how methods of complex analysis can be used to give new results on the conformal Calabi-Yau problem concerning the existence of bounded metrically complete minimal surfaces in real Euclidean 3-space R^3. We shall see in particular that every bordered Riemann surface admits a proper complete holomorphic immersion into the ball of C^2, and a proper complete embedding as a holomorphic null curve into the ball of C^3. Since the real and the imaginary parts of a holomorphic null curve in C^3 are conformally immersed minimal surfaces in R^3, we obtain a bounded complete conformal minimal immersion of any bordered Riemann surface into R^3. The main advantage of our methods, when compared to the existing ones in the literature, is that we do not need to change the conformal type of the Riemann surface. (Joint work with A. Alarcon, University of Granada.)
Hormander's estimate, some generalizations and new applications
12:10 Mon 17 Feb, 2014 :: Ingkarni Wardli B20 :: Prof Zbigniew Blocki :: Jagiellonian University

Lars Hormander proved his estimate for the d-bar equation in 1965. It is one the most important results in several complex variables (SCV). New applications have emerged recently, outside of SCV. We will present three of them: the Ohsawa-Takegoshi extension theorem with optimal constant, the one-dimensional Suita Conjecture, and Nazarov's approach to the Bourgain-Milman inequality from convex analysis.
The structuring role of chaotic stirring on pelagic ecosystems
11:10 Fri 28 Feb, 2014 :: B19 Ingkarni Wardli :: Dr Francesco d'Ovidio :: Universite Pierre et Marie Curie (Paris VI)

The open ocean upper layer is characterized by a complex transport dynamics occuring over different spatiotemporal scales. At the scale of 10-100 km - which covers the so called mesoscale and part of the submesoscale - in situ and remote sensing observations detect strong variability in physical and biogeochemical fields like sea surface temperature, salinity, and chlorophyll concentration. The calculation of Lyapunov exponent and other nonlinear diagnostics applied to the surface currents have allowed to show that an important part of this tracer variability is due to chaotic stirring. Here I will extend this analysis to marine ecosystems. For primary producers, I will show that stable and unstable manifolds of hyperbolic points embedded in the surface velocity field are able to structure the phytoplanktonic community in fluid dynamical niches of dominant types, where competition can locally occur during bloom events. By using data from tagged whales, frigatebirds, and elephant seals, I will also show that chaotic stirring affects the behaviour of higher trophic levels. In perspective, these relations between transport structures and marine ecosystems can be the base for a biodiversity index constructued from satellite information, and therefore able to monitor key aspects of the marine biodiversity and its temporal variability at the global scale.
The effects of pre-existing immunity
15:10 Fri 7 Mar, 2014 :: B.18 Ingkarni Wardli :: Associate Professor Jane Heffernan :: York University, Canada

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Immune system memory, also called immunity, is gained as a result of primary infection or vaccination, and can be boosted after vaccination or secondary infections. Immunity is developed so that the immune system is primed to react and fight a pathogen earlier and more effectively in secondary infections. The effects of memory, however, on pathogen propagation in an individual host (in-host) and a population (epidemiology) are not well understood. Mathematical models of infectious diseases, employing dynamical systems, computer simulation and bifurcation analysis, can provide projections of pathogen propagation, show outcomes of infection and help inform public health interventions. In the Modelling Infection and Immunity (MI^2) lab, we develop and study biologically informed mathematical models of infectious diseases at both levels of infection, and combine these models into comprehensive multi-scale models so that the effects of individual immunity in a population can be determined. In this talk we will discuss some of the interesting mathematical phenomenon that arise in our models, and show how our results are directly applicable to what is known about the persistence of infectious diseases.
Dynamical systems approach to fluid-plasma turbulence
15:10 Fri 14 Mar, 2014 :: 5.58 Ingkarni Wardli :: Professor Abraham Chian

Sun-Earth system is a complex, electrodynamically coupled system dominated by multiscale interactions. The complex behavior of the space environment is indicative of a state driven far from equilibrium whereby instabilities, nonlinear waves, and turbulence play key roles in the system dynamics. First, we review the fundamental concepts of nonlinear dynamics in fluids and plasmas and discuss their relevance to the study of the Sun-Earth relation. Next, we show how Lagrangian coherent structures identify the transport barriers of plasma turbulence modeled by 3-D solar convective dynamo. Finally, we show how Lagrangian coherent structures can be detected in the solar photospheric turbulence using satellite observations.
Embed to homogenise heterogeneous wave equation.
12:35 Mon 17 Mar, 2014 :: B.19 Ingkarni Wardli :: Chen Chen :: University of Adelaide

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Consider materials with complicated microstructure: we want to model their large scale dynamics by equations with effective, `average' coefficients. I will show an example of heterogeneous wave equation in 1D. If Centre manifold theory is applied to model the original heterogeneous wave equation directly, we will get a trivial model. I embed the wave equation into a family of more complex wave problems and I show the equivalence of the two sets of solutions.
Moduli spaces of contact instantons
12:10 Fri 28 Mar, 2014 :: Ingkarni Wardli B20 :: David Baraglia :: University of Adelaide

In dimensions greater than four there are several notions of higher Yang-Mills instantons. This talk concerns one such case, contact instantons, defined for 5-dimensional contact manifolds. The geometry transverse to the Reeb foliation turns out to be important in understanding the moduli space. For example, we show the dimension of the moduli space is the index of a transverse elliptic complex. This is joint work with Pedram Hekmati.
A model for the BitCoin block chain that takes propagation delays into account
15:10 Fri 28 Mar, 2014 :: B.21 Ingkarni Wardli :: Professor Peter Taylor :: The University of Melbourne

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Unlike cash transactions, most electronic transactions require the presence of a trusted authority to verify that the payer has sufficient funding to be able to make the transaction and to adjust the account balances of the payer and payee. In recent years BitCoin has been proposed as an "electronic equivalent of cash". The general idea is that transactions are verified in a coded form in a block chain, which is maintained by the community of participants. Problems can arise when the block chain splits: that is different participants have different versions of the block chain, something which can happen only when there are propagation delays, at least if all participants are behaving according to the protocol. In this talk I shall present a preliminary model for the splitting behaviour of the block chain. I shall then go on to perform a similar analysis for a situation where a group of participants has adopted a recently-proposed strategy for gaining a greater advantage from BitCoin processing than its combined computer power should be able to control.
Semiclassical restriction estimates
12:10 Fri 4 Apr, 2014 :: Ingkarni Wardli B20 :: Melissa Tacy :: University of Adelaide

Eigenfunctions of Hamiltonians arise naturally in the theory of quantum mechanics as stationary states of quantum systems. Their eigenvalues have an interpretation as the square root of E, where E is the energy of the system. We wish to better understand the high energy limit which defines the boundary between quantum and classical mechanics. In this talk I will focus on results regarding the restriction of eigenfunctions to lower dimensional subspaces, in particular to hypersurfaces. A convenient way to study such problems is to reframe them as problems in semiclassical analysis.
Bayesian Indirect Inference
12:10 Mon 14 Apr, 2014 :: B.19 Ingkarni Wardli :: Brock Hermans :: University of Adelaide

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Bayesian likelihood-free methods saw the resurgence of Bayesian statistics through the use of computer sampling techniques. Since the resurgence, attention has focused on so-called 'summary statistics', that is, ways of summarising data that allow for accurate inference to be performed. However, it is not uncommon to find data sets in which the summary statistic approach is not sufficient. In this talk, I will be summarising some of the likelihood-free methods most commonly used (don't worry if you've never seen any Bayesian analysis before), as well as looking at Bayesian Indirect Likelihood, a new way of implementing Bayesian analysis which combines new inference methods with some of the older computational algorithms.
Network-based approaches to classification and biomarker identification in metastatic melanoma
15:10 Fri 2 May, 2014 :: B.21 Ingkarni Wardli :: Associate Professor Jean Yee Hwa Yang :: The University of Sydney

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Finding prognostic markers has been a central question in much of current research in medicine and biology. In the last decade, approaches to prognostic prediction within a genomics setting are primarily based on changes in individual genes / protein. Very recently, however, network based approaches to prognostic prediction have begun to emerge which utilize interaction information between genes. This is based on the believe that large-scale molecular interaction networks are dynamic in nature and changes in these networks, rather than changes in individual genes/proteins, are often drivers of complex diseases such as cancer. In this talk, I use data from stage III melanoma patients provided by Prof. Mann from Melanoma Institute of Australia to discuss how network information can be utilize in the analysis of gene expression analysis to aid in biological interpretation. Here, we explore a number of novel and previously published network-based prediction methods, which we will then compare to the common single-gene and gene-set methods with the aim of identifying more biologically interpretable biomarkers in the form of networks.
The Mandelbrot Set
12:10 Mon 5 May, 2014 :: B.19 Ingkarni Wardli :: David Bowman :: University of Adelaide

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The Mandelbrot set is an icon of modern mathematics, an image which fires the popular imagination when accompanied by the words 'chaos' and 'fractal'. However, few could give even a vague definition of this mysterious set and fewer still know the mathematical meaning behind it. In this talk we will be looking at the role that the Mandelbrot set plays in complex dynamics, the study of iterated complex valued functions. We shall discuss attracting and repelling cycles and how they are related to the different components of the Mandelbrot set.
Multiple Sclerosis and linear stability analysis
12:35 Mon 19 May, 2014 :: B.19 Ingkarni Wardli :: Saber Dini :: University of Adelaide

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Multiple sclerosis (MS), is an inflammatory disease in which the immune system of the body attacks the myelin sheaths around axons in the brain and damages, or in other words, demyelinates the axons. Demyelination process can lead to scarring as well as a broad spectrum of signs and symptoms. Brain of vertebrates has a mechanism to restore the demyelination or Remyelinate the damaged area. Remyelination in the brain is accomplished by glial cells (servers of neurons). Glial cells should accumulate in the damaged areas of the brain to start the repairing process and this accumulation can be viewed as instability. Therefore, spatiotemporal linear stability analysis can be undertaken on the issue to investigate quantitative aspects of the remyelination process.
Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space
12:10 Fri 6 Jun, 2014 :: Ingkarni Wardli B20 :: Finnur Larusson :: University of Adelaide

I will discuss new joint work with Franc Forstneric. The group of holomorphic automorphisms of complex affine space C^n, n>1, is huge. It is not an infinite-dimensional manifold in any recognised sense. Still, our work shows that in some ways it behaves like a finite-dimensional Oka manifold.
Group meeting
15:10 Fri 6 Jun, 2014 :: 5.58 Ingkarni Wardli :: Meng Cao and Trent Mattner :: University of Adelaide

Meng Cao:: Multiscale modelling couples patches of nonlinear wave-like simulations :: Abstract: The multiscale gap-tooth scheme is built from given microscale simulations of complicated physical processes to empower macroscale simulations. By coupling small patches of simulations over unsimulated physical gaps, large savings in computational time are possible. So far the gap-tooth scheme has been developed for dissipative systems, but wave systems are also of great interest. This article develops the gap-tooth scheme to the case of nonlinear microscale simulations of wave-like systems. Classic macroscale interpolation provides a generic coupling between patches that achieves arbitrarily high order consistency between the multiscale scheme and the underlying microscale dynamics. Eigen-analysis indicates that the resultant gap-tooth scheme empowers feasible computation of large scale simulations of wave-like dynamics with complicated underlying physics. As an pilot study, we implement numerical simulations of dam-breaking waves by the gap-tooth scheme. Comparison between a gap-tooth simulation, a microscale simulation over the whole domain, and some published experimental data on dam breaking, demonstrates that the gap-tooth scheme feasibly computes large scale wave-like dynamics with computational savings. Trent Mattner :: Coupled atmosphere-fire simulations of the Canberra 2003 bushfires using WRF-Sfire :: Abstract: The Canberra fires of January 18, 2003 are notorious for the extreme fire behaviour and fire-atmosphere-topography interactions that occurred, including lee-slope fire channelling, pyrocumulonimbus development and tornado formation. In this talk, I will discuss coupled fire-weather simulations of the Canberra fires using WRF-SFire. In these simulations, a fire-behaviour model is used to dynamically predict the evolution of the fire front according to local atmospheric and topographic conditions, as well as the associated heat and moisture fluxes to the atmosphere. It is found that the predicted fire front and heat flux is not too bad, bearing in mind the complexity of the problem and the severe modelling assumptions made. However, the predicted moisture flux is too low, which has some impact on atmospheric dynamics.
Complexifications, Realifications, Real forms and Complex Structures
12:10 Mon 23 Jun, 2014 :: B.19 Ingkarni Wardli :: Kelli Francis-Staite :: University of Adelaide

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Italian mathematicians Niccolò Fontana Tartaglia and Gerolamo Cardano introduced complex numbers to solve polynomial equations such as x^2+1=0. Solving a standard real differential equation often uses complex eigenvalues and eigenfunctions. In both cases, the solution space is expanded to include the complex numbers, solved, and then translated back to the real case. My talk aims to explain the process of complexification and related concepts. It will give vocabulary and some basic results about this important process. And it will contain cute cat pictures.
Hydrodynamics and rheology of self-propelled colloids
15:10 Fri 8 Aug, 2014 :: B17 Ingkarni Wardli :: Dr Sarthok Sircar :: University of Adelaide

The sub-cellular world has many components in common with soft condensed matter systems (polymers, colloids and liquid crystals). But it has novel properties, not present in traditional complex fluids, arising from a rich spectrum of non-equilibrium behavior: flocking, chemotaxis and bioconvection. The talk is divided into two parts. In the first half, we will (get an idea on how to) derive a hydrodynamic model for self-propelled particles of an arbitrary shape from first principles, in a sufficiently dilute suspension limit, moving in a 3-dimensional space inside a viscous solvent. The model is then restricted to particles with ellipsoidal geometry to quantify the interplay of the long-range excluded volume and the short-range self-propulsion effects. The expression for the constitutive stresses, relating the kinetic theory with the momentum transport equations, are derived using a combination of the virtual work principle (for extra elastic stresses) and symmetry arguments (for active stresses). The second half of the talk will highlight on my current numerical expertise. In particular we will exploit a specific class of spectral basis functions together with RK4 time-stepping to determine the dynamical phases/structures as well as phase-transitions of these ellipsoidal clusters. We will also discuss on how to define the order (or orientation) of these clusters and understand the other rheological quantities.
T-duality and the chiral de Rham complex
12:10 Fri 22 Aug, 2014 :: Ingkarni Wardli B20 :: Andrew Linshaw :: University of Denver

The chiral de Rham complex of Malikov, Schechtman, and Vaintrob is a sheaf of vertex algebras that exists on any smooth manifold M. It has a square-zero differential D, and contains the algebra of differential forms on M as a subcomplex. In this talk, I'll give an introduction to vertex algebras and sketch this construction. Finally, I'll discuss a notion of T-duality in this setting. This is based on joint work in progress with V. Mathai.
Software and protocol verification using Alloy
12:10 Mon 25 Aug, 2014 :: B.19 Ingkarni Wardli :: Dinesha Ranathunga :: University of Adelaide

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Reliable software isn't achieved by trial and error. It requires tools to support verification. Alloy is a tool based on set theory that allows expression of a logic-based model of software or a protocol, and hence allows checking of this model. In this talk, I will cover its key concepts, language syntax and analysis features.
Ideal membership on singular varieties by means of residue currents
12:10 Fri 29 Aug, 2014 :: Ingkarni Wardli B20 :: Richard Larkang :: University of Adelaide

On a complex manifold X, one can consider the following ideal membership problem: Does a holomorphic function on X belong to a given ideal of holomorphic functions on X? Residue currents give a way of expressing analytically this essentially algebraic problem. I will discuss some basic cases of this, why such an analytic description might be useful, and finish by discussing a generalization of this to singular varieties.
Neural Development of the Visual System: a laminar approach
15:10 Fri 29 Aug, 2014 :: N132 Engineering North :: Dr Andrew Oster :: Eastern Washington University

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In this talk, we will introduce the architecture of the visual system in higher order primates and cats. Through activity-dependent plasticity mechanisms, the left and right eye streams segregate in the cortex in a stripe-like manner, resulting in a pattern called an ocular dominance map. We introduce a mathematical model to study how such a neural wiring pattern emerges. We go on to consider the joint development of the ocular dominance map with another feature of the visual system, the cytochrome oxidase blobs, which appear in the center of the ocular dominance stripes. Since cortex is in fact comprised of layers, we introduce a simple laminar model and perform a stability analysis of the wiring pattern. This intricate biological structure (ocular dominance stripes with "blobs" periodically distributed in their centers) can be understood as occurring due to two Turing instabilities combined with the leading-order dynamics of the system.
Neural Development of the Visual System: a laminar approach
15:10 Fri 29 Aug, 2014 :: This talk will now be given as a School Colloquium :: Dr Andrew Oster :: Eastern Washington University

In this talk, we will introduce the architecture of the visual system in higher order primates and cats. Through activity-dependent plasticity mechanisms, the left and right eye streams segregate in the cortex in a stripe-like manner, resulting in a pattern called an ocular dominance map. We introduce a mathematical model to study how such a neural wiring pattern emerges. We go on to consider the joint development of the ocular dominance map with another feature of the visual system, the cytochrome oxidase blobs, which appear in the center of the ocular dominance stripes. Since cortex is in fact comprised of layers, we introduce a simple laminar model and perform a stability analysis of the wiring pattern. This intricate biological structure (ocular dominance stripes with 'blobs' periodically distributed in their centers) can be understood as occurring due to two Turing instabilities combined with the leading-order dynamics of the system.
Modelling biological gel mechanics
12:10 Mon 8 Sep, 2014 :: B.19 Ingkarni Wardli :: James Reoch :: University of Adelaide

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The behaviour of gels such as collagen is the result of complex interactions between mechanical and chemical forces. In this talk, I will outline the modelling approaches we are looking at in order to incorporate the influence of cell behaviour alongside chemical potentials, and the various circumstances which lead to gel swelling and contraction.
Inferring absolute population and recruitment of southern rock lobster using only catch and effort data
12:35 Mon 22 Sep, 2014 :: B.19 Ingkarni Wardli :: John Feenstra :: University of Adelaide

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Abundance estimates from a data-limited version of catch survey analysis are compared to those from a novel one-parameter deterministic method. Bias of both methods is explored using simulation testing based on a more complex data-rich stock assessment population dynamics fishery operating model, exploring the impact of both varying levels of observation error in data as well as model process error. Recruitment was consistently better estimated than legal size population, the latter most sensitive to increasing observation errors. A hybrid of the data-limited methods is proposed as the most robust approach. A more statistically conventional error-in-variables approach may also be touched upon if enough time.
To Complex Analysis... and beyond!
12:10 Mon 29 Sep, 2014 :: B.19 Ingkarni Wardli :: Brett Chenoweth :: University of Adelaide

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In the undergraduate complex analysis course students learn about complex valued functions on domains in C (the complex plane). Several interesting and surprising results come about from this study. In my talk I will introduce a more general setting where complex analysis can be done, namely Riemann surfaces (complex manifolds of dimension 1). I will then prove that all non-compact Riemann surfaces are Stein; which loosely speaking means that their function theory is similar to that of C.
The Serre-Grothendieck theorem by geometric means
12:10 Fri 24 Oct, 2014 :: Ingkarni Wardli B20 :: David Roberts :: University of Adelaide

The Serre-Grothendieck theorem implies that every torsion integral 3rd cohomology class on a finite CW-complex is the invariant of some projective bundle. It was originally proved in a letter by Serre, used homotopical methods, most notably a Postnikov decomposition of a certain classifying space with divisible homotopy groups. In this talk I will outline, using work of the algebraic geometer Offer Gabber, a proof for compact smooth manifolds using geometric means and a little K-theory.
Modelling segregation distortion in multi-parent crosses
15:00 Mon 17 Nov, 2014 :: 5.57 Ingkarni Wardli :: Rohan Shah (joint work with B. Emma Huang and Colin R. Cavanagh) :: The University of Queensland

Construction of high-density genetic maps has been made feasible by low-cost high-throughput genotyping technology; however, the process is still complicated by biological, statistical and computational issues. A major challenge is the presence of segregation distortion, which can be caused by selection, difference in fitness, or suppression of recombination due to introgressed segments from other species. Alien introgressions are common in major crop species, where they have often been used to introduce beneficial genes from wild relatives. Segregation distortion causes problems at many stages of the map construction process, including assignment to linkage groups and estimation of recombination fractions. This can result in incorrect ordering and estimation of map distances. While discarding markers will improve the resulting map, it may result in the loss of genomic regions under selection or containing beneficial genes (in the case of introgression). To correct for segregation distortion we model it explicitly in the estimation of recombination fractions. Previously proposed methods introduce additional parameters to model the distortion, with a corresponding increase in computing requirements. This poses difficulties for large, densely genotyped experimental populations. We propose a method imposing minimal additional computational burden which is suitable for high-density map construction in large multi-parent crosses. We demonstrate its use modelling the known Sr36 introgression in wheat for an eight-parent complex cross.
Nonlinear analysis over infinite dimensional spaces and its applications
12:10 Fri 6 Feb, 2015 :: Ingkarni Wardli B20 :: Tsuyoshi Kato :: Kyoto University

In this talk we develop moduli theory of holomorphic curves over infinite dimensional manifolds consisted by sequences of almost Kaehler manifolds. Under the assumption of high symmetry, we verify that many mechanisms of the standard moduli theory over closed symplectic manifolds also work over these infinite dimensional spaces. As an application, we study deformation theory of discrete groups acting on trees. There is a canonical way, up to conjugacy to embed such groups into the automorphism group over the infinite projective space. We verify that for some class of Hamiltonian functions, the deformed groups must be always asymptotically infinite.
On the analyticity of CR-diffeomorphisms
12:10 Fri 13 Mar, 2015 :: Engineering North N132 :: Ilya Kossivskiy :: University of Vienna

One of the fundamental objects in several complex variables is CR-mappings. CR-mappings naturally occur in complex analysis as boundary values of mappings between domains, and as restrictions of holomorphic mappings onto real submanifolds. It was already observed by Cartan that smooth CR-diffeomorphisms between CR-submanifolds in C^N tend to be very regular, i.e., they are restrictions of holomorphic maps. However, in general smooth CR-mappings form a more restrictive class of mappings. Thus, since the inception of CR-geometry, the following general question has been of fundamental importance for the field: Are CR-equivalent real-analytic CR-structures also equivalent holomorphically? In joint work with Lamel, we answer this question in the negative, in any positive CR-dimension and CR-codimension. Our construction is based on a recent dynamical technique in CR-geometry, developed in my earlier work with Shafikov.
Groups acting on trees
12:10 Fri 10 Apr, 2015 :: Napier 144 :: Anitha Thillaisundaram :: Heinrich Heine University of Duesseldorf

From a geometric point of view, branch groups are groups acting spherically transitively on a spherically homogeneous rooted tree. The applications of branch groups reach out to analysis, geometry, combinatorics, and probability. The early construction of branch groups were the Grigorchuk group and the Gupta-Sidki p-groups. Among its many claims to fame, the Grigorchuk group was the first example of a group of intermediate growth (i.e. neither polynomial nor exponential). Here we consider a generalisation of the family of Grigorchuk-Gupta-Sidki groups, and we examine the restricted occurrence of their maximal subgroups.
A Collision Algorithm for Sea Ice
12:10 Mon 4 May, 2015 :: Napier LG29 :: Lucas Yiew :: University of Adelaide

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The wave-induced collisions between sea ice are highly complex and nonlinear, and involves a multitude of subprocesses. Several collision models do exist, however, to date, none of these models have been successfully integrated into sea-ice forecasting models. A key component of a collision model is the development of an appropriate collision algorithm. In this seminar I will present a time-stepping, event-driven algorithm to detect, analyse and implement the pre- and post-collision processes.
Some approaches toward a stronger Jacobian conjecture
12:10 Fri 5 Jun, 2015 :: Napier 144 :: Tuyen Truong :: University of Adelaide

The Jacobian conjecture states that if a polynomial self-map of C^n has invertible Jacobian, then the map has a polynomial inverse. Is it true, false or simply undecidable? In this talk I will propose a conjecture concerning general square matrices with complex coefficients, whose validity implies the Jacobian conjecture. The conjecture is checked in various cases, in particular it is true for generic matrices. Also, a heuristic argument is provided explaining why the conjecture (and thus, also the Jacobian conjecture) should be true.
Complex Systems, Chaotic Dynamics and Infectious Diseases
15:10 Fri 5 Jun, 2015 :: Engineering North N132 :: Prof Michael Small :: UWA

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In complex systems, the interconnection between the components of the system determine the dynamics. The system is described by a very large and random mathematical graph and it is the topological structure of that graph which is important for understanding of the dynamical behaviour of the system. I will talk about two specific examples - (1) spread of infectious disease (where the connection between the agents in a population, rather than epidemic parameters, determine the endemic state); and, (2) a transformation to represent a dynamical system as a graph (such that the "statistical mechanics" of the graph characterise the dynamics).
Workshop on Geometric Quantisation
10:10 Mon 27 Jul, 2015 :: Level 7 conference room Ingkarni Wardli :: Michele Vergne, Weiping Zhang, Eckhard Meinrenken, Nigel Higson and many others

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Geometric quantisation has been an increasingly active area since before the 1980s, with links to physics, symplectic geometry, representation theory, index theory, and differential geometry and geometric analysis in general. In addition to its relevance as a field on its own, it acts as a focal point for the interaction between all of these areas, which has yielded far-reaching and powerful results. This workshop features a large number of international speakers, who are all well-known for their work in (differential) geometry, representation theory and/or geometric analysis. This is a great opportunity for anyone interested in these areas to meet and learn from some of the top mathematicians in the world. Students are especially welcome. Registration is free.
Dynamics on Networks: The role of local dynamics and global networks on hypersynchronous neural activity
15:10 Fri 31 Jul, 2015 :: Ingkarni Wardli B21 :: Prof John Terry :: University of Exeter, UK

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Graph theory has evolved into a useful tool for studying complex brain networks inferred from a variety of measures of neural activity, including fMRI, DTI, MEG and EEG. In the study of neurological disorders, recent work has discovered differences in the structure of graphs inferred from patient and control cohorts. However, most of these studies pursue a purely observational approach; identifying correlations between properties of graphs and the cohort which they describe, without consideration of the underlying mechanisms. To move beyond this necessitates the development of mathematical modelling approaches to appropriately interpret network interactions and the alterations in brain dynamics they permit.

In the talk we introduce some of these concepts with application to epilepsy, introducing a dynamic network approach to study resting state EEG recordings from a cohort of 35 people with epilepsy and 40 adult controls. Using this framework we demonstrate a strongly significant difference between networks inferred from the background activity of people with epilepsy in comparison to normal controls. Our findings demonstrate that a mathematical model based analysis of routine clinical EEG provides significant additional information beyond standard clinical interpretation, which may ultimately enable a more appropriate mechanistic stratification of people with epilepsy leading to improved diagnostics and therapeutics.

Mathematical Modeling and Analysis of Active Suspensions
14:10 Mon 3 Aug, 2015 :: Napier 209 :: Professor Michael Shelley :: Courant Institute of Mathematical Sciences, New York University

Complex fluids that have a 'bio-active' microstructure, like suspensions of swimming bacteria or assemblies of immersed biopolymers and motor-proteins, are important examples of so-called active matter. These internally driven fluids can have strange mechanical properties, and show persistent activity-driven flows and self-organization. I will show how first-principles PDE models are derived through reciprocal coupling of the 'active stresses' generated by collective microscopic activity to the fluid's macroscopic flows. These PDEs have an interesting analytic structures and dynamics that agree qualitatively with experimental observations: they predict the transitions to flow instability and persistent mixing observed in bacterial suspensions, and for microtubule assemblies show the generation, propagation, and annihilation of disclination defects. I'll discuss how these models might be used to study yet more complex biophysical systems.
In vitro models of colorectal cancer: why and how?
15:10 Fri 7 Aug, 2015 :: B19 Ingkarni Wardli :: Dr Tamsin Lannagan :: Gastrointestinal Cancer Biology Group, University of Adelaide / SAHMRI

1 in 20 Australians will develop colorectal cancer (CRC) and it is the second most common cause of cancer death. Similar to many other cancer types, it is the metastases rather than the primary tumour that are lethal, and prognosis is defined by “how far” the tumour has spread at time of diagnosis. Modelling in vivo behavior through rapid and relatively inexpensive in vitro assays would help better target therapies as well as help develop new treatments. One such new in vitro tool is the culture of 3D organoids. Organoids are a biologically stable means of growing, storing and testing treatments against bowel cancer. To this end, we have just set up a human colorectal organoid bank across Australia. This consortium will help us to relate in vitro growth patterns to in vivo behaviour and ultimately in the selection of patients for personalized therapies. Organoid growth, however, is complex. There appears to be variable growth rates and growth patterns. Together with members of the ECMS we recently gained funding to better quantify and model spatial structures in these colorectal organoids. This partnership will aim to directly apply the expertise within the ECMS to patient care.
Deformation retractions from the space of continuous maps between domains in C onto the space of holomorphic maps
12:10 Mon 17 Aug, 2015 :: Benham Labs G10 :: Brett Chenoweth :: University of Adelaide

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Mikhail Gromov proved in 1989 that every continuous map from a Stein manifold S to an elliptic manifold X could be deformed to a holomorphic map. More generally, it is true that if X is an Oka manifold then a continuous map from a Stein source into X can always be deformed to a holomorphic map. The question is whether we can do this for all continuous maps at once, in a `nice' way that does not change a map f if f is already holomorphic. In a recent paper by Larusson, we see that ANRs play an important in producing a partial answer to this question. In this talk we will explore the question in the relatively simple situation where the source and target are domains in the complex plane.
Pattern Formation in Nature
12:10 Mon 31 Aug, 2015 :: Benham Labs G10 :: Saber Dini :: University of Adelaide

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Pattern formation is a ubiquitous process in nature: embryo development, animals skin pigmentation, etc. I will talk about how Alan Turing (the British genius known for the Turing Machine) explained pattern formation by linear stability analysis of reaction-diffusion systems.
T-duality and bulk-boundary correspondence
12:10 Fri 11 Sep, 2015 :: Ingkarni Wardli B17 :: Guo Chuan Thiang :: The University of Adelaide

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Bulk-boundary correspondences in physics can be modelled as topological boundary homomorphisms in K-theory, associated to an extension of a "bulk algebra" by a "boundary algebra". In joint work with V. Mathai, such bulk-boundary maps are shown to T-dualize into simple restriction maps in a large number of cases, generalizing what the Fourier transform does for ordinary functions. I will give examples, involving both complex and real K-theory, and explain how these results may be used to study topological phases of matter and D-brane charges in string theory.
Base change and K-theory
12:10 Fri 18 Sep, 2015 :: Ingkarni Wardli B17 :: Hang Wang :: The University of Adelaide

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Tempered representations of an algebraic group can be classified by K-theory of the corresponding group C^*-algebra. We use Archimedean base change between Langlands parameters of real and complex algebraic groups to compare K-theory of the corresponding C^*-algebras of groups over different number fields. This is work in progress with K.F. Chao.
Real Lie Groups and Complex Flag Manifolds
12:10 Fri 9 Oct, 2015 :: Ingkarni Wardli B17 :: Joseph A. Wolf :: University of California, Berkeley

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Let G be a complex simple direct limit group. Let G_R be a real form of G that corresponds to an hermitian symmetric space. I'll describe the corresponding bounded symmetric domain in the context of the Borel embedding, Cayley transforms, and the Bergman-Shilov boundary. Let Q be a parabolic subgroup of G. In finite dimensions this means that G/Q is a complex projective variety, or equivalently has a Kaehler metric invariant under a maximal compact subgroup of G. Then I'll show just how the bounded symmetric domains describe cycle spaces for open G_R orbits on G/Q. These cycle spaces include the complex bounded symmetric domains. In finite dimensions they are tightly related to moduli spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite dimensions there are more problems than answers. Finally, time permitting, I'll indicate how some of this goes over to real and to quaternionic bounded symmetric domains.
IGA/AMSI Workshop -- Australia-Japan Geometry, Analysis and their Applications
09:00 Mon 19 Oct, 2015 :: Ingkarni Wardli Conference Room 7.15 (Level 7)

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Interdisciplinary workshop between Australia and Japan on Geometry, Analysis and their Applications.
Oka principles and the linearization problem
12:10 Fri 8 Jan, 2016 :: Engineering North N132 :: Gerald Schwarz :: Brandeis University

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Let G be a reductive complex Lie group (e.g., SL(n,C)) and let X and Y be Stein manifolds (closed complex submanifolds of some C^n). Suppose that G acts freely on X and Y. Then there are quotient Stein manifolds X/G and Y/G and quotient mappings p_X:X-> X/G and p_Y: Y-> Y/G such that X and Y are principal G-bundles over X/G and Y/G. Let us suppose that Q=X/G ~= Y/G so that X and Y have the same quotient Q. A map Phi: X\to Y of principal bundles (over Q) is simply an equivariant continuous map commuting with the projections. That is, Phi(gx)=g Phi(x) for all g in G and x in X, and p_X=p_Y o Phi. The famous Oka Principle of Grauert says that any Phi as above embeds in a continuous family Phi_t: X -> Y, t in [0,1], where Phi_0=Phi, all the Phi_t satisfy the same conditions as Phi does and Phi_1 is holomorphic. This is rather amazing. We consider the case where G does not necessarily act freely on X and Y. There is still a notion of quotient and quotient mappings p_X: X-> X//G and p_Y: Y-> Y//G where X//G and Y//G are now Stein spaces and parameterize the closed G-orbits in X and Y. We assume that Q~= X//G~= Y//G and that we have a continuous equivariant Phi such that p_X=p_Y o Phi. We find conditions under which Phi embeds into a continuous family Phi_t such that Phi_1 is holomorphic. We give an application to the Linearization Problem. Let G act holomorphically on C^n. When is there a biholomorphic map Phi:C^n -> C^n such that Phi^{-1} o g o Phi in GL(n,C) for all g in G? We find a condition which is necessary and sufficient for "most" G-actions. This is joint work with F. Kutzschebauch and F. Larusson.
A fibered density property and the automorphism group of the spectral ball
12:10 Fri 15 Jan, 2016 :: Engineering North N132 :: Frank Kutzschebauch :: University of Bern

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The spectral ball is defined as the set of complex n by n matrices whose eigenvalues are all less than 1 in absolute value. Its group of holomorphic automorphisms has been studied over many decades in several papers and a precise conjecture about its structure has been formulated. In dimension 2 this conjecture was recently disproved by Kosinski. We not only disprove the conjecture in all dimensions but also give the best possible description of the automorphism group. Namely we explain how the invariant theoretic quotient map divides the automorphism group of the spectral ball into a finite dimensional part of symmetries which lift from the quotient and an infinite dimensional part which leaves the fibration invariant. We prove a precise statement as to how hopelessly huge this latter part is. This is joint work with R. Andrist.
A long C^2 without holomorphic functions
12:10 Fri 29 Jan, 2016 :: Engineering North N132 :: Franc Forstneric :: University of Ljubljana

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For every integer n>1 we construct a complex manifold of dimension n which is exhausted by an increasing sequence of biholomorphic images of C^n (i.e., a long C^n), but it does not admit any nonconstant holomorphic functions. We also introduce new biholomorphic invariants of a complex manifold, the stable core and the strongly stable core, and we prove that every compact strongly pseudoconvex and polynomially convex domain B in C^n is the strongly stable core of a long C^n; in particular, non-equivalent domains give rise to non-equivalent long C^n's. Thus, for any n>1 there exist uncountably many pairwise non-equivalent long C^n's. These results answer several long standing open questions. (Joint work with Luka Boc Thaler.)
T-duality for elliptic curve orientifolds
12:10 Fri 4 Mar, 2016 :: Ingkarni Wardli B17 :: Jonathan Rosenberg :: University of Maryland

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Orientifold string theories are quantum field theories based on the geometry of a space with an involution. T-dualities are certain relationships between such theories that look different on the surface but give rise to the same observable physics. In this talk I will not assume any knowledge of physics but will concentrate on the associated geometry, in the case where the underlying space is a (complex) elliptic curve and the involution is either holomorphic or anti-holomorphic. The results blend algebraic topology and algebraic geometry. This is mostly joint work with Chuck Doran and Stefan Mendez-Diez.
How predictable are you? Information and happiness in social media.
12:10 Mon 21 Mar, 2016 :: Ingkarni Wardli Conference Room 715 :: Dr Lewis Mitchell :: School of Mathematical Sciences

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The explosion of ``Big Data'' coming from online social networks and the like has opened up the new field of ``computational social science'', which applies a quantitative lens to problems traditionally in the domain of psychologists, anthropologists and social scientists. What does it mean to be influential? How do ideas propagate amongst populations? Is happiness contagious? For the first time, mathematicians, statisticians, and computer scientists can provide insight into these and other questions. Using data from social networks such as Facebook and Twitter, I will give an overview of recent research trends in computational social science, describe some of my own work using techniques like sentiment analysis and information theory in this realm, and explain how you can get involved with this highly rewarding research field as well.
Geometric analysis of gap-labelling
12:10 Fri 8 Apr, 2016 :: Eng & Maths EM205 :: Mathai Varghese :: University of Adelaide

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Using an earlier result, joint with Quillen, I will formulate a gap labelling conjecture for magnetic Schrodinger operators with smooth aperiodic potentials on Euclidean space. Results in low dimensions will be given, and the formulation of the same problem for certain non-Euclidean spaces will be given if time permits. This is ongoing joint work with Moulay Benameur.
Sard Theorem for the endpoint map in sub-Riemannian manifolds
12:10 Fri 29 Apr, 2016 :: Eng & Maths EM205 :: Alessandro Ottazzi :: University of New South Wales

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Sub-Riemannian geometries occur in several areas of pure and applied mathematics, including harmonic analysis, PDEs, control theory, metric geometry, geometric group theory, and neurobiology. We introduce sub-Riemannian manifolds and give some examples. Therefore we discuss some of the open problems, and in particular we focus on the Sard Theorem for the endpoint map, which is related to the study of length minimizers. Finally, we consider some recent results obtained in collaboration with E. Le Donne, R. Montgomery, P. Pansu and D. Vittone.
How to count Betti numbers
12:10 Fri 6 May, 2016 :: Eng & Maths EM205 :: David Baraglia :: University of Adelaide

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I will begin this talk by showing how to obtain the Betti numbers of certain smooth complex projective varieties by counting points over a finite field. For singular or non-compact varieties this motivates us to consider the "virtual Hodge numbers" encoded by the "Hodge-Deligne polynomial", a refinement of the topological Euler characteristic. I will then discuss the computation of Hodge-Deligne polynomials for certain singular character varieties (i.e. moduli spaces of flat connections).
Harmonic analysis of Hodge-Dirac operators
12:10 Fri 13 May, 2016 :: Eng & Maths EM205 :: Pierre Portal :: Australian National University

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When the metric on a Riemannian manifold is perturbed in a rough (merely bounded and measurable) manner, do basic estimates involving the Hodge Dirac operator $D = d+d^*$ remain valid? Even in the model case of a perturbation of the euclidean metric on $\mathbb{R}^n$, this is a difficult question. For instance, the fact that the $L^2$ estimate $\|Du\|_2 \sim \|\sqrt{D^{2}}u\|_2$ remains valid for perturbed versions of $D$ was a famous conjecture made by Kato in 1961 and solved, positively, in a ground breaking paper of Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002. In the past fifteen years, a theory has emerged from the solution of this conjecture, making rough perturbation problems much more tractable. In this talk, I will give a general introduction to this theory, and present one of its latest results: a flexible approach to $L^p$ estimates for the holomorphic functional calculus of $D$. This is joint work with D. Frey (Delft) and A. McIntosh (ANU).
Harmonic Analysis in Rough Contexts
15:10 Fri 13 May, 2016 :: Engineering South S112 :: Dr Pierre Portal :: Australian National University

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In recent years, perspectives on what constitutes the ``natural" framework within which to conduct various forms of mathematical analysis have shifted substantially. The common theme of these shifts can be described as a move towards roughness, i.e. the elimination of smoothness assumptions that had previously been considered fundamental. Examples include partial differential equations on domains with a boundary that is merely Lipschitz continuous, geometric analysis on metric measure spaces that do not have a smooth structure, and stochastic analysis of dynamical systems that have nowhere differentiable trajectories. In this talk, aimed at a general mathematical audience, I describe some of these shifts towards roughness, placing an emphasis on harmonic analysis, and on my own contributions. This includes the development of heat kernel methods in situations where such a kernel is merely a distribution, and applications to deterministic and stochastic partial differential equations.
Smooth mapping orbifolds
12:10 Fri 20 May, 2016 :: Eng & Maths EM205 :: David Roberts :: University of Adelaide

It is well-known that orbifolds can be represented by a special kind of Lie groupoid, namely those that are étale and proper. Lie groupoids themselves are one way of presenting certain nice differentiable stacks. In joint work with Ray Vozzo we have constructed a presentation of the mapping stack Hom(disc(M),X), for M a compact manifold and X a differentiable stack, by a Fréchet-Lie groupoid. This uses an apparently new result in global analysis about the map C^\infty(K_1,Y) \to C^\infty(K_2,Y) induced by restriction along the inclusion K_2 \to K_1, for certain compact K_1,K_2. We apply this to the case of X being an orbifold to show that the mapping stack is an infinite-dimensional orbifold groupoid. We also present results about mapping groupoids for bundle gerbes.
Time series analysis of paleo-climate proxies (a mathematical perspective)
15:10 Fri 27 May, 2016 :: Engineering South S112 :: Dr Thomas Stemler :: University of Western Australia

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In this talk I will present the work my colleagues from the School of Earth and Environment (UWA), the "trans disciplinary methods" group of the Potsdam Institute for Climate Impact Research, Germany, and I did to explain the dynamics of the Australian-South East Asian monsoon system during the last couple of thousand years. From a time series perspective paleo-climate proxy series are more or less the monsters moving under your bed that wake you up in the middle of the night. The data is clearly non-stationary, non-uniform sampled in time and the influence of stochastic forcing or the level of measurement noise are more or less unknown. Given these undesirable properties almost all traditional time series analysis methods fail. I will highlight two methods that allow us to draw useful conclusions from the data sets. The first one uses Gaussian kernel methods to reconstruct climate networks from multiple proxies. The coupling relationships in these networks change over time and therefore can be used to infer which areas of the monsoon system dominate the complex dynamics of the whole system. Secondly I will introduce the transformation cost time series method, which allows us to detect changes in the dynamics of a non-uniform sampled time series. Unlike the frequently used interpolation approach, our new method does not corrupt the data and therefore avoids biases in any subsequence analysis. While I will again focus on paleo-climate proxies, the method can be used in other applied areas, where regular sampling is not possible.
Multi-scale modeling in biofluids and particle aggregation
15:10 Fri 17 Jun, 2016 :: B17 Ingkarni Wardli :: Dr Sarthok Sircar :: University of Adelaide

In today's seminar I will give 2 examples in mathematical biology which describes the multi-scale organization at 2 levels: the meso/micro level and the continuum/macro level. I will then detail suitable tools in statistical mechanics to link these different scales. The first problem arises in mathematical physiology: swelling-de-swelling mechanism of mucus, an ionic gel. Mucus is packaged inside cells at high concentration (volume fraction) and when released into the extracellular environment, it expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is due to the rapid exchange of calcium and sodium that changes the cross-linked structure of the mucus polymers, thereby causing it to swell. Modeling this problem involves a two-phase, polymer/solvent mixture theory (in the continuum level description), together with the chemistry of the polymer, its nearest neighbor interaction and its binding with the dissolved ionic species (in the micro-scale description). The problem is posed as a free-boundary problem, with the boundary conditions derived from a combination of variational principle and perturbation analysis. The dynamics of neutral gels and the equilibrium-states of the ionic gels are analyzed. In the second example, we numerically study the adhesion fragmentation dynamics of rigid, round particles clusters subject to a homogeneous shear flow. In the macro level we describe the dynamics of the number density of these cluster. The description in the micro-scale includes (a) binding/unbinding of the bonds attached on the particle surface, (b) bond torsion, (c) surface potential due to ionic medium, and (d) flow hydrodynamics due to shear flow.
Chern-Simons invariants of Seifert manifolds via Loop spaces
14:10 Tue 28 Jun, 2016 :: Ingkarni Wardli B17 :: Ryan Mickler :: Northeastern University

Over the past 30 years the Chern-Simons functional for connections on G-bundles over three-manfolds has lead to a deep understanding of the geometry of three-manfiolds, as well as knot invariants such as the Jones polynomial. Here we study this functional for three-manfolds that are topologically given as the total space of a principal circle bundle over a compact Riemann surface base, which are known as Seifert manifolds. We show that on such manifolds the Chern-Simons functional reduces to a particular gauge-theoretic functional on the 2d base, that describes a gauge theory of connections on an infinite dimensional bundle over this base with structure group given by the level-k affine central extension of the loop group LG. We show that this formulation gives a new understanding of results of Beasley-Witten on the computability of quantum Chern-Simons invariants of these manifolds as well as knot invariants for knots that wrap a single fiber of the circle bundle. A central tool in our analysis is the Caloron correspondence of Murray-Stevenson-Vozzo.
Holomorphic Flexibility Properties of Spaces of Elliptic Functions
12:10 Fri 29 Jul, 2016 :: Ingkarni Wardli B18 :: David Bowman :: University of Adelaide

The set of meromorphic functions on an elliptic curve naturally possesses the structure of a complex manifold. The component of degree 3 functions is 6-dimensional and enjoys several interesting complex-analytic properties that make it, loosely speaking, the opposite of a hyperbolic manifold. Our main result is that this component has a 54-sheeted branched covering space that is an Oka manifold.
Approaches to modelling cells and remodelling biological tissues
14:10 Wed 10 Aug, 2016 :: Ingkarni Wardli 5.57 :: Professor Helen Byrne :: University of Oxford

Biological tissues are complex structures, whose evolution is characterised by multiple biophysical processes that act across diverse space and time scales. For example, during normal wound healing, fibroblast cells located around the wound margin exert contractile forces to close the wound while those located in the surrounding tissue synthesise new tissue in response to local growth factors and mechanical stress created by wound contraction. In this talk I will illustrate how mathematical modelling can provide insight into such complex processes, taking my inspiration from recent studies of cell migration, vasculogenesis and wound healing.
Calculus on symplectic manifolds
12:10 Fri 12 Aug, 2016 :: Ingkarni Wardli B18 :: Mike Eastwood :: University of Adelaide

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One can use the symplectic form to construct an elliptic complex replacing the de Rham complex. Then, under suitable curvature conditions, one can form coupled versions of this complex. Finally, on complex projective space, these constructions give rise to a series of elliptic complexes with geometric consequences for the Fubini-Study metric and its X-ray transform. This talk, which will start from scratch, is based on the work of many authors but, especially, current joint work with Jan Slovak.
Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type
12:10 Fri 19 Aug, 2016 :: Ingkarni Wardli B18 :: Lesley Ward :: University of South Australia

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Much effort has been devoted to generalizing the Calder'on-Zygmund theory in harmonic analysis from Euclidean spaces to metric measure spaces, or spaces of homogeneous type. Here the underlying space R^n with Euclidean metric and Lebesgue measure is replaced by a set X with general metric or quasi-metric and a doubling measure. Further, one can replace the Laplacian operator that underpins the Calderon-Zygmund theory by more general operators L satisfying heat kernel estimates. I will present recent joint work with P. Chen, X.T. Duong, J. Li and L.X. Yan along these lines. We develop the theory of product Hardy spaces H^p_{L_1,L_2}(X_1 x X_2), for 1
A principled experimental design approach to big data analysis
15:10 Fri 23 Sep, 2016 :: Napier G03 :: Prof Kerrie Mengersen :: Queensland University of Technology

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Big Datasets are endemic, but they are often notoriously difficult to analyse because of their size, complexity, history and quality. The purpose of this paper is to open a discourse on the use of modern experimental design methods to analyse Big Data in order to answer particular questions of interest. By appeal to a range of examples, it is suggested that this perspective on Big Data modelling and analysis has wide generality and advantageous inferential and computational properties. In particular, the principled experimental design approach is shown to provide a flexible framework for analysis that, for certain classes of objectives and utility functions, delivers equivalent answers compared with analyses of the full dataset. It can also provide a formalised method for iterative parameter estimation, model checking, identification of data gaps and evaluation of data quality. Finally it has the potential to add value to other Big Data sampling algorithms, in particular divide-and-conquer strategies, by determining efficient sub-samples.
SIR epidemics with stages of infection
12:10 Wed 28 Sep, 2016 :: EM218 :: Matthieu Simon :: Universite Libre de Bruxelles

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This talk is concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population. The population is subdivided into three classes of individuals: the susceptibles, the infectives and the removed cases. In short, an infective remains infectious during a random period of time. While infected, it can contact all the susceptibles present, independently of the other infectives. At the end of the infectious period, it becomes a removed case and has no further part in the infection process.

We represent an infectious period as a set of different stages that an infective can go through before being removed. The transitions between stages are ruled by either a Markov process or a semi-Markov process. In each stage, an infective makes contaminations at the epochs of a Poisson process with a specific rate.

Our purpose is to derive closed expressions for a transform of different statistics related to the end of the epidemic, such as the final number of susceptibles and the area under the trajectories of all the infectives. The analysis is performed by using simple matrix analytic methods and martingale arguments. Numerical illustrations will be provided at the end of the talk.
Parahoric bundles, invariant theory and the Kazhdan-Lusztig map
12:10 Fri 21 Oct, 2016 :: Ingkarni Wardli B18 :: David Baraglia :: University of Adelaide

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In this talk I will introduce the notion of parahoric groups, a loop group analogue of parabolic subgroups. I will also discuss a global version of this, namely parahoric bundles on a complex curve. This leads us to a problem concerning the behaviour of invariant polynomials on the dual of the Lie algebra, a kind of "parahoric invariant theory". The key to solving this problem turns out to be the Kazhdan-Lusztig map, which assigns to each nilpotent orbit in a semisimple Lie algebra a conjugacy class in the Weyl group. Based on joint work with Masoud Kamgarpour and Rohith Varma.
Leavitt path algebras
12:10 Fri 2 Dec, 2016 :: Engineering & Math EM213 :: Roozbeh Hazrat :: Western Sydney University

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From a directed graph one can generate an algebra which captures the movements along the graph. One such algebras are Leavitt path algebras. Despite being introduced only 10 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, a theory which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered. In this talk, we introduce Leavitt path algebras and try to classify them by means of (graded) Grothendieck groups. We will ask nice questions!
An equivariant parametric Oka principle for bundles of homogeneous spaces
12:10 Fri 3 Mar, 2017 :: Napier 209 :: Finnur Larusson :: University of Adelaide

I will report on new joint work with Frank Kutzschebauch and Gerald Schwarz (arXiv:1612.07372). Under certain conditions, every continuous section of a holomorphic fibre bundle can be deformed to a holomorphic section. In fact, the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. What if a complex Lie group acts on the bundle and its sections? We have proved an analogous result for equivariant sections. The result has a wide scope. If time permits, I will describe some interesting special cases and mention two applications.
Collective and aneural foraging in biological systems
15:10 Fri 3 Mar, 2017 :: Lower Napier LG14 :: Dr Jerome Buhl and Dr David Vogel :: The University of Adelaide

The field of collective behaviour uses concepts originally adapted from statistical physics to study how complex collective phenomena such as mass movement or swarm intelligence emerge from relatively simple interactions between individuals. Here we will focus on two applications of this framework. First we will have look at new insights into the evolution of sociality brought by combining models of nutrition and social interactions to explore phenomena such as collective foraging decisions, emergence of social organisation and social immunity. Second, we will look at the networks built by slime molds under exploration and foraging context.
What is index theory?
12:10 Tue 21 Mar, 2017 :: Inkgarni Wardli 5.57 :: Dr Peter Hochs :: School of Mathematical Sciences

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Index theory is a link between topology, geometry and analysis. A typical theorem in index theory says that two numbers are equal: an analytic index and a topological index. The first theorem of this kind was the index theorem of Atiyah and Singer, which they proved in 1963. Index theorems have many applications in maths and physics. For example, they can be used to prove that a differential equation must have a solution. Also, they imply that the topology of a space like a sphere or a torus determines in what ways it can be curved. Topology is the study of geometric properties that do not change if we stretch or compress a shape without cutting or glueing. Curvature does change when we stretch something out, so it is surprising that topology can say anything about curvature. Index theory has many surprising consequences like this.
Minimal surfaces and complex analysis
12:10 Fri 24 Mar, 2017 :: Napier 209 :: Antonio Alarcon :: University of Granada

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A surface in the Euclidean space R^3 is said to be minimal if it is locally area-minimizing, meaning that every point in the surface admits a compact neighborhood with the least area among all the surfaces with the same boundary. Although the origin of minimal surfaces is in physics, since they can be realized locally as soap films, this family of surfaces lies in the intersection of many fields of mathematics. In particular, complex analysis in one and several variables plays a fundamental role in the theory. In this lecture we will discuss the influence of complex analysis in the study of minimal surfaces.
Geometric structures on moduli spaces
12:10 Fri 31 Mar, 2017 :: Napier 209 :: Nicholas Buchdahl :: University of Adelaide

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Moduli spaces are used to classify various kinds of objects, often arising from solutions of certain differential equations on manifolds; for example, the complex structures on a compact surface or the anti-self-dual Yang-Mills equations on an oriented smooth 4-manifold. Sometimes these moduli spaces carry important information about the underlying manifold, manifested most clearly in the results of Donaldson and others on the topology of smooth 4-manifolds. It is also the case that these moduli spaces themselves carry interesting geometric structures; for example, the Weil-Petersson metric on moduli spaces of compact Riemann surfaces, exploited to great effect by Maryam Mirzakhani. In this talk, I shall elaborate on the theme of geometric structures on moduli spaces, with particular focus on some recent-ish work done in conjunction with Georg Schumacher.
K-types of tempered representations
12:10 Fri 7 Apr, 2017 :: Napier 209 :: Peter Hochs :: University of Adelaide

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Tempered representations of a reductive Lie group G are the irreducible unitary representations one needs in the Plancherel decomposition of L^2(G). They are relevant to harmonic analysis because of this, and also occur in the Langlands classification of the larger class of admissible representations. If K in G is a maximal compact subgroup, then there is a considerable amount of information in the restriction of a tempered representation to K. In joint work with Yanli Song and Shilin Yu, we give a geometric expression for the decomposition of such a restriction into irreducibles. The multiplicities of these irreducibles are expressed as indices of Dirac operators on reduced spaces of a coadjoint orbit of G corresponding to the representation. These reduced spaces are Spin-c analogues of reduced spaces in symplectic geometry, defined in terms of moment maps that represent conserved quantities. This result involves a Spin-c version of the quantisation commutes with reduction principle for noncompact manifolds. For discrete series representations, this was done by Paradan in 2003.
Schubert Calculus on Lagrangian Grassmannians
12:10 Tue 23 May, 2017 :: EM 213 :: Hiep Tuan Dang :: National centre for theoretical sciences, Taiwan

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The Lagrangian Grassmannian $LG = LG(n,2n)$ is the projective complex manifold which parametrizes Lagrangian (i.e. maximal isotropic) subspaces in a symplective vector space of dimension $2n$. This talk is mainly devoted to Schubert calculus on $LG$. We first recall the definition of Schubert classes in this context. Then we present basic results which are similar to the classical formulas due to Pieri and Giambelli. These lead to a presentation of the cohomology ring of $LG$. Finally, we will discuss recent results related to the Schubert structure constants and Gromov-Witten invariants of $LG$.
Holomorphic Legendrian curves
12:10 Fri 26 May, 2017 :: Napier 209 :: Franc Forstneric :: University of Ljubljana, Slovenia

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I will present recent results on the existence and behaviour of noncompact holomorphic Legendrian curves in complex contact manifolds. We show that these curves are ubiquitous in \C^{2n+1} with the standard holomorphic contact form \alpha=dz+\sum_{j=1}^n x_jdy_j; in particular, every open Riemann surface embeds into \C^3 as a proper holomorphic Legendrian curves. On the other hand, for any integer n>= 1 there exist Kobayashi hyperbolic complex contact structures on \C^{2n+1} which do not admit any nonconstant Legendrian complex lines. Furthermore, we construct a holomorphic Darboux chart around any noncompact holomorphic Legendrian curve in an arbitrary complex contact manifold. As an application, we show that every bordered holomorphic Legendrian curve can be uniformly approximated by complete bounded Legendrian curves.
Complex methods in real integral geometry
12:10 Fri 28 Jul, 2017 :: Engineering Sth S111 :: Mike Eastwood :: University of Adelaide

There are well-known analogies between holomorphic integral transforms such as the Penrose transform and real integral transforms such as the Radon, Funk, and John transforms. In fact, one can make a precise connection between them and hence use complex methods to establish results in the real setting. This talk will introduce some simple integral transforms and indicate how complex analysis may be applied.
Weil's Riemann hypothesis (RH) and dynamical systems
12:10 Fri 11 Aug, 2017 :: Engineering Sth S111 :: Tuyen Truong :: University of Adelaide

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Weil proposed an analogue of the RH in finite fields, aiming at counting asymptotically the number of solutions to a given system of polynomial equations (with coefficients in a finite field) in finite field extensions of the base field. This conjecture influenced the development of Algebraic Geometry since the 1950’s, most important achievements include: Grothendieck et al.’s etale cohomology, and Bombieri and Grothendieck’s standard conjectures on algebraic cycles (inspired by a Kahlerian analogue of a generalisation of Weil’s RH by Serre). Weil’s RH was solved by Deligne in the 70’s, but the finite field analogue of Serre’s result is still open (even in dimension 2). This talk presents my recent work proposing a generalisation of Weil’s RH by relating it to standard conjectures and a relatively new notion in complex dynamical systems called dynamical degrees. In the course of the talk, I will present the proof of a question proposed by Esnault and Srinivas (which is related to a result by Gromov and Yomdin on entropy of complex dynamical systems), which gives support to the finite field analogue of Serre’s result.
Mathematics is Biology's Next Microscope (Only Better!)
15:10 Fri 11 Aug, 2017 :: Ingkarni Wardli B17 :: Dr Robyn Araujo :: Queensland University of Technology

While mathematics has long been considered "an essential tool for physics", the foundations of biology and the life sciences have received significantly less influence from mathematical ideas and theory. In this talk, I will give a brief discussion of my recent research on robustness in molecular signalling networks, as an example of a complex biological question that calls for a mathematical answer. In particular, it has been a long-standing mystery how the extraordinarily complex communication networks inside living cells, comprising thousands of different interacting molecules, are able to function robustly since complexity is generally associated with fragility. Mathematics has now suggested a resolution to this paradox through the discovery that robust adaptive signalling networks must be constructed from a just small number of well-defined universal modules (or "motifs"), connected together. The existence of these newly-discovered modules has important implications for evolutionary biology, embryology and development, cancer research, and drug development.
Mathematics is Biology'€™s Next Microscope (Only Better!)
15:10 Fri 11 Aug, 2017 :: Ingkarni Wardli B17 :: Dr Robyn Araujo :: Queensland University of Technology

While mathematics has long been considered “an essential tool for physics", the foundations of biology and the life sciences have received significantly less influence from mathematical ideas and theory. In this talk, I will give a brief discussion of my recent research on robustness in molecular signalling networks, as an example of a complex biological question that calls for a mathematical answer. In particular, it has been a long-standing mystery how the extraordinarily complex communication networks inside living cells, comprising thousands of different interacting molecules, are able to function robustly since complexity is generally associated with fragility. Mathematics has now suggested a resolution to this paradox through the discovery that robust adaptive signalling networks must be constructed from a just small number of well-defined universal modules (or “motifs”), connected together. The existence of these newly-discovered modules has important implications for evolutionary biology, embryology and development, cancer research, and drug development.
Dynamics of transcendental Hanon maps
11:10 Wed 20 Sep, 2017 :: Engineering & Math EM212 :: Leandro Arosio :: University of Rome

The dynamics of a polynomial in the complex plane is a classical topic studied already at the beginning of the 20th century by Fatou and Julia. The complex plane is partitioned in two natural invariant sets: a compact set called the Julia set, with (usually) fractal structure and chaotic behaviour, and the Fatou set, where dynamics has no sensitive dependence on initial conditions. The dynamics of a transcendental map was first studied by Baker fifty years ago, and shows striking differences with the polynomial case: for example, there are wandering Fatou components. Moving to C^2, an analogue of polynomial dynamics is given by Hanon maps, polynomial automorphisms with interesting dynamics. In this talk I will introduce a natural generalisation of transcendental dynamics to C^2, and show how to construct wandering domains for such maps.
On directions and operators
11:10 Wed 27 Sep, 2017 :: Engineering & Math EM213 :: Malabika Pramanik :: University of British Columbia

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Many fundamental operators arising in harmonic analysis are governed by sets of directions that they are naturally associated with. This talk will survey a few representative results in this area, and report on some new developments.
Stochastic Modelling of Urban Structure
11:10 Mon 20 Nov, 2017 :: Engineering Nth N132 :: Mark Girolami :: Imperial College London, and The Alan Turing Institute

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Urban systems are complex in nature and comprise of a large number of individuals that act according to utility, a measure of net benefit pertaining to preferences. The actions of individuals give rise to an emergent behaviour, creating the so-called urban structure that we observe. In this talk, I develop a stochastic model of urban structure to formally account for uncertainty arising from the complex behaviour. We further use this stochastic model to infer the components of a utility function from observed urban structure. This is a more powerful modelling framework in comparison to the ubiquitous discrete choice models that are of limited use for complex systems, in which the overall preferences of individuals are difficult to ascertain. We model urban structure as a realization of a Boltzmann distribution that is the invariant distribution of a related stochastic differential equation (SDE) that describes the dynamics of the urban system. Our specification of Boltzmann distribution assigns higher probability to stable configurations, in the sense that consumer surplus (demand) is balanced with running costs (supply), as characterized by a potential function. We specify a Bayesian hierarchical model to infer the components of a utility function from observed structure. Our model is doubly-intractable and poses significant computational challenges that we overcome using recent advances in Markov chain Monte Carlo (MCMC) methods. We demonstrate our methodology with case studies on the London retail system and airports in England.
Calculating optimal limits for transacting credit card customers
15:10 Fri 2 Mar, 2018 :: Horace Lamb 1022 :: Prof Peter Taylor :: University of Melbourne

Credit card users can roughly be divided into `transactors', who pay off their balance each month, and `revolvers', who maintain an outstanding balance, on which they pay substantial interest. In this talk, we focus on modelling the behaviour of an individual transactor customer. Our motivation is to calculate an optimal credit limit from the bank's point of view. This requires an expression for the expected outstanding balance at the end of a payment period. We establish a connection with the classical newsvendor model. Furthermore, we derive the Laplace transform of the outstanding balance, assuming that purchases are made according to a marked point process and that there is a simplified balance control policy which prevents all purchases in the rest of the payment period when the credit limit is exceeded. We then use the newsvendor model and our modified model to calculate bounds on the optimal credit limit for the more realistic balance control policy that accepts all purchases that do not exceed the limit. We illustrate our analysis using a compound Poisson process example and show that the optimal limit scales with the distribution of the purchasing process, while the probability of exceeding the optimal limit remains constant. Finally, we apply our model to some real credit card purchase data.
Radial Toeplitz operators on bounded symmetric domains
11:10 Fri 9 Mar, 2018 :: Lower Napier LG11 :: Raul Quiroga-Barranco :: CIMAT, Guanajuato, Mexico

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The Bergman spaces on a complex domain are defined as the space of holomorphic square-integrable functions on the domain. These carry interesting structures both for analysis and representation theory in the case of bounded symmetric domains. On the other hand, these spaces have some bounded operators obtained as the composition of a multiplier operator and a projection. These operators are highly noncommuting between each other. However, there exist large commutative C*-algebras generated by some of these Toeplitz operators very much related to Lie groups. I will construct an example of such C*-algebras and provide a fairly explicit simultaneous diagonalization of the generating Toeplitz operators.
Chaos in higher-dimensional complex dynamics
13:10 Fri 20 Apr, 2018 :: Barr Smith South Polygon Lecture theatre :: Finnur Larusson :: University of Adelaide

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I will report on new joint work with Leandro Arosio (University of Rome, Tor Vergata). Complex manifolds can be thought of as laid out across a spectrum characterised by rigidity at one end and flexibility at the other. On the rigid side, Kobayashi-hyperbolic manifolds have at most a finite-dimensional group of symmetries. On the flexible side, there are manifolds with an extremely large group of holomorphic automorphisms, the prototypes being the affine spaces $\mathbb C^n$ for $n \geq 2$. From a dynamical point of view, hyperbolicity does not permit chaos. An endomorphism of a Kobayashi-hyperbolic manifold is non-expansive with respect to the Kobayashi distance, so every family of endomorphisms is equicontinuous. We show that not only does flexibility allow chaos: under a strong anti-hyperbolicity assumption, chaotic automorphisms are generic. A special case of our main result is that if $G$ is a connected complex linear algebraic group of dimension at least 2, not semisimple, then chaotic automorphisms are generic among all holomorphic automorphisms of $G$ that preserve a left- or right-invariant Haar form. For $G=\mathbb C^n$, this result was proved (although not explicitly stated) some 20 years ago by Fornaess and Sibony. Our generalisation follows their approach. I will give plenty of context and background, as well as some details of the proof of the main result.
The topology and geometry of spaces of Yang-Mills-Higgs flow lines
11:10 Fri 27 Jul, 2018 :: Barr Smith South Polygon Lecture theatre :: Graeme Wilkin :: National University of Singapore

Given a smooth complex vector bundle over a compact Riemann surface, one can define the space of Higgs bundles and an energy functional on this space: the Yang-Mills-Higgs functional. The gradient flow of this functional resembles a nonlinear heat equation, and the limit of the flow detects information about the algebraic structure of the initial Higgs bundle (e.g. whether or not it is semistable). In this talk I will explain my work to classify ancient solutions of the Yang-Mills-Higgs flow in terms of their algebraic structure, which leads to an algebro-geometric classification of Yang-Mills-Higgs flow lines. Critical points connected by flow lines can then be interpreted in terms of the Hecke correspondence, which appears in Witten’s recent work on Geometric Langlands. This classification also gives a geometric description of spaces of unbroken flow lines in terms of secant varieties of the underlying Riemann surface, and in the remaining time I will describe work in progress to relate the (analytic) Morse compactification of these spaces by broken flow lines to an algebro-geometric compactification by iterated blowups of secant varieties.
Equivariant Index, Traces and Representation Theory
11:10 Fri 10 Aug, 2018 :: Barr Smith South Polygon Lecture theatre :: Hang Wang :: University of Adelaide

K-theory of C*-algebras associated to a semisimple Lie group can be understood both from the geometric point of view via Baum-Connes assembly map and from the representation theoretic point of view via harmonic analysis of Lie groups. A K-theory generator can be viewed as the equivariant index of some Dirac operator, but also interpreted as a (family of) representation(s) parametrised by the noncompact abelian part in the Levi component of a cuspidal parabolic subgroup. Applying orbital traces to the K-theory group, we obtain the equivariant index as a fixed point formula which, for each K-theory generators for (limit of) discrete series, recovers Harish-Chandra’s character formula on the representation theory side. This is a noncompact analogue of Atiyah-Segal-Singer fixed point theorem in relation to the Weyl character formula. This is joint work with Peter Hochs.
Projected Particle Filters
15:10 Fri 24 Aug, 2018 :: Lower Napier LG15 :: Dr John Maclean :: University of Adelaide

cientific advances owe equally to models and data, and both will remain relevant and key to further understanding. Observations drive model development, and model development often drives data acquisition. It therefore is particularly prudent to have these two sides of the scientific coin work in concert. This is a mathematical and statistical question: how to combine the output of model investigations and observational data. The area that is dedicated to studying and developing the best approaches to this issue is called Data Assimilation (DA). Perhaps the most crucial modern-day application of DA is numerical weather prediction, but it is also used in GPS systems and studies of atmospheric conditions on other planets. I will take the probabilistic or Bayesian approach to DA. At a particular time at which data are available, the question of data assimilation is how to approximate the posterior or analysis distribution, that is found by conditioning the "forecast distribution" on the data. A key method under this umbrella is the particle filter, that approximates the forecast and posterior distributions with an ensemble of weighted particles. The talk will focus on a contribution to particle filtering made from a dynamical systems point of view. I will introduce a framework for Particle Filtering, PF-AUS, in which only the components of data corresponding to the unstable and neutral modes of the forecast model are assimilated. The particle filter is well suited to nonlinear forecast models, and non-Gaussian forecast distributions, but would normally require exponentially more computational effort as the dimension of the DA problem increases. The PF-AUS implementation is shown to correspond to assimilating observations of a lower dimension, equal to the number of Lyapunov exponents. The dimension of the observations is crucial in the computational cost of the particle filter and this approach is a framework to drastically lower that cost while preserving as much relevant information as possible, in that the unstable and neutral modes correspond to the most uncertain model predictions. Particle filters are an active area of research in both the DA and the statistical communities, and there are many competing algorithms. One nice feature of PF-AUS is that it is not exactly an algorithm but rather a framework for filtering: any particle filter can be applied in the PF-AUS framework.
Topological Data Analysis
15:10 Fri 31 Aug, 2018 :: Napier 208 :: Dr Vanessa Robins :: Australian National University

Topological Data Analysis has grown out of work focussed on deriving qualitative and yet quantifiable information about the shape of data. The underlying assumption is that knowledge of shape - the way the data are distributed - permits high-level reasoning and modelling of the processes that created this data. The 0-th order aspect of shape is the number pieces: "connected components" to a topologist; "clustering" to a statistician. Higher-order topological aspects of shape are holes, quantified as "non-bounding cycles" in homology theory. These signal the existence of some type of constraint on the data-generating process. Homology lends itself naturally to computer implementation, but its naive application is not robust to noise. This inspired the development of persistent homology: an algebraic topological tool that measures changes in the topology of a growing sequence of spaces (a filtration). Persistent homology provides invariants called the barcodes or persistence diagrams that are sets of intervals recording the birth and death parameter values of each homology class in the filtration. It captures information about the shape of data over a range of length scales, and enables the identification of "noisy" topological structure. Statistical analysis of persistent homology has been challenging because the raw information (the persistence diagrams) are provided as sets of intervals rather than functions. Various approaches to converting persistence diagrams to functional forms have been developed recently, and have found application to data ranging from the distribution of galaxies, to porous materials, and cancer detection.

News matching "Complex analysis"

Workshop on Complex Geometry
The Institute for Geometry and its Applications will host a Workshop on Complex Geometry at the University of Adelaide from Monday 16 February to Friday 20 February 2009. Click here for full details. Posted Wed 17 Sep 08.
ARC Grant successes
The School of Mathematical Sciences has again had outstanding success in the ARC Discovery and Linkage Projects schemes. Congratulations to the following staff for their success in the Discovery Project scheme: Prof Nigel Bean, Dr Josh Ross, Prof Phil Pollett, Prof Peter Taylor, New methods for improving active adaptive management in biological systems, $255,000 over 3 years; Dr Josh Ross, New methods for integrating population structure and stochasticity into models of disease dynamics, $248,000 over three years; A/Prof Matt Roughan, Dr Walter Willinger, Internet traffic-matrix synthesis, $290,000 over three years; Prof Patricia Solomon, A/Prof John Moran, Statistical methods for the analysis of critical care data, with application to the Australian and New Zealand Intensive Care Database, $310,000 over 3 years; Prof Mathai Varghese, Prof Peter Bouwknegt, Supersymmetric quantum field theory, topology and duality, $375,000 over 3 years; Prof Peter Taylor, Prof Nigel Bean, Dr Sophie Hautphenne, Dr Mark Fackrell, Dr Malgorzata O'Reilly, Prof Guy Latouche, Advanced matrix-analytic methods with applications, $600,000 over 3 years. Congratulations to the following staff for their success in the Linkage Project scheme: Prof Simon Beecham, Prof Lee White, A/Prof John Boland, Prof Phil Howlett, Dr Yvonne Stokes, Mr John Wells, Paving the way: an experimental approach to the mathematical modelling and design of permeable pavements, $370,000 over 3 years; Dr Amie Albrecht, Prof Phil Howlett, Dr Andrew Metcalfe, Dr Peter Pudney, Prof Roderick Smith, Saving energy on trains - demonstration, evaluation, integration, $540,000 over 3 years Posted Fri 29 Oct 10.
New Fellow of the Australian Academy of Science
Professor Mathai Varghese, Professor of Pure Mathematics and ARC Professorial Fellow within the School of Mathematical Sciences, was elected to the Australian Academy of Science. Professor Varghese's citation read "for his distinguished for his work in geometric analysis involving the topology of manifolds, including the Mathai-Quillen formalism in topological field theory.". Posted Tue 30 Nov 10.
ARC Grant Success
Congratulations to the following staff who were successful in securing funding from the Australian Research Council Discovery Projects Scheme. Associate Professor Finnur Larusson awarded $270,000 for his project Flexibility and symmetry in complex geometry; Dr Thomas Leistner, awarded $303,464 for his project Holonomy groups in Lorentzian geometry, Professor Michael Murray Murray and Dr Daniel Stevenson (Glasgow), awarded $270,000 for their project Bundle gerbes: generalisations and applications; Professor Mathai Varghese, awarded $105,000 for his project Advances in index theory and Prof Anthony Roberts and Professor Ioannis Kevrekidis (Princeton) awarded $330,000 for their project Accurate modelling of large multiscale dynamical systems for engineering and scientific simulation and analysis Posted Tue 8 Nov 11.
Elder Professor Mathai Varghese Awarded Australian Laureate Fellowship
Professor Mathai Varghese, Elder Professor of Mathematics in the School of Mathematical Sciences, has been awarded an Australian Laureate Fellowship worth $1.64 million to advance Index Theory and its applications. The project is expected to enhance Australia’s position at the forefront of international research in geometric analysis. Posted Thu 15 Jun 17.

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Elder Professor Mathai Varghese Awarded Australian Laureate Fellowship
Professor Mathai Varghese, Elder Professor of Mathematics in the School of Mathematical Sciences, has been awarded an Australian Laureate Fellowship worth $1.64 million to advance Index Theory and its applications. The project will enhance Australia's position at the forefront of international research in geometric analysis. Posted Thu 15 Jun 17.

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Publications matching "Complex analysis"

Publications
Inversion of analytically perturbed linear operators that are singular at the origin
Howlett, P; Avrachenkov, K; Pearce, Charles; Ejov, V, Journal of Mathematical Analysis and Applications 353 (68–84) 2009
Portfolio risk minimization and differential games
Elliott, Robert; Siu, T, Nonlinear Analysis-Theory Methods & Applications In Press (–) 2009
Schlicht Envelopes of Holomorphy and Foliations by Lines
Larusson, Finnur; Shafikov, R, Journal of Geometric Analysis 19 (373–389) 2009
A total probability approach to flood frequency analysis in tidal river reaches
Need, Steven; Lambert, Martin; Metcalfe, Andrew, World Environmental and Water Resources Congress 2008 Ahupua'a, Honolulu 12/05/08
Quantitative analysis ofincorrectly-configured bogon-filter detection
Arnold, Jonathan; Maennel, Olaf; Flavel, Ashley; McMahon, Jeremy; Roughan, Matthew, Australasian Telecommunication Networks and Applications Conference, Adelaide 07/12/08
A non-linear filter
Elliott, Robert; Leung, H; Deng, J, Stochastic Analysis and Applications 26 (856–862) 2008
Frequency analysis of rainfall and streamflow extremes accounting for seasonal and climatic partitions
Leonard, Michael; Metcalfe, Andrew; Lambert, Martin, Journal of Hydrology 348 (135–147) 2008
Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM
Azis, Mohammad; Clements, David, Engineering Analysis With Boundary Elements 32 (1054–1060) 2008
Internet traffic and multiresolution analysis
Zhang, Y; Ge, Z; Diggavi, S; Mao, Z; Roughan, Matthew; Vaishampayan, V; Willinger, W; Zhang, Y, chapter in Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Institute of Mathematical Statistic) 215–234, 2008
Aspects of Dirac operators in analysis
Eastwood, Michael; Ryan, J, Milan Journal of Mathematics 75 (91–116) 2007
Gene expression analysis of multiple gastrointestinal regions reveals activation of common cell regulatory pathways following cytotoxic chemotherapy
Bowen, Joanne; Gibson, Rachel; Tsykin, Anna; Stringer, Andrea Marie; Logan, Richard; Keefe, Dorothy, International Journal of Cancer 121 (1847–1856) 2007
Nonclassical symmetry solutions for reaction-diffusion equations with explicity spatial dependence
Hajek, Bronwyn; Edwards, M; Broadbridge, P; Williams, G, Nonlinear Analysis-Theory Methods & Applications 67 (2541–2552) 2007
Optimal multilinear estimation of a random vector under constraints of casualty and limited memory
Howlett, P; Torokhti, Anatoli; Pearce, Charles, Computational Statistics & Data Analysis 52 (869–878) 2007
Statistics in review; Part 2: Generalised linear models, time-to-event and time-series analysis, evidence synthesis and clinical trials
Moran, John; Solomon, Patricia, Critical care and Resuscitation 9 (187–197) 2007
The solution of a free boundary problem related to environmental management systems
Elliott, Robert; Filinkov, Alexei, Stochastic Analysis and Applications 25 (1189–1202) 2007
Experimental Design and Analysis of Microarray Data
Wilson, C; Tsykin, Anna; Wilkinson, Christopher; Abbott, C, chapter in Bioinformatics (Elsevier Ltd) 1–36, 2006
A Markov analysis of social learning and adaptation
Wheeler, Scott; Bean, Nigel; Gaffney, Janice; Taylor, Peter, Journal of Evolutionary Economics 16 (299–319) 2006
Data-recursive smoother formulae for partially observed discrete-time Markov chains
Elliott, Robert; Malcolm, William, Stochastic Analysis and Applications 24 (579–597) 2006
Mathematical analysis of an extended mumford-shah model for image segmentation
Tao, Trevor; Crisp, David; Van Der Hoek, John, Journal of Mathematical Imaging and Vision 24 (327–340) 2006
Methodology in meta-analysis: a study from critical care meta-analytic practice
Moran, John; Solomon, Patricia; Warn, D, Health Services and Outcomes Research Methodology 5 (207–226) 2006
On the indentation of an inhomogeneous anisotropic elastic material by multiple straight rigid punches
Clements, David; Ang, W, Engineering Analysis With Boundary Elements 30 (284–291) 2006
Some Penrose transforms in complex differential geometry
Anco, S; Bland, J; Eastwood, Michael, Science in China Series A-Mathematics Physics Astronomy 49 (1599–1610) 2006
Stochastic volatility model with filtering
Elliott, Robert; MIao, H, Stochastic Analysis and Applications 24 (661–683) 2006
The influence of urban land-use on non-motorised transport casualties
Wedagama, D; Bird, R; Metcalfe, Andrew, Accident Analysis and Prevention 38 (1049–1057) 2006
Three-dimensional flow due to a microcantilever oscillating near a wall: an unsteady slender-body analysis
Clarke, Richard; Jensen, O; Billingham, J; Williams, P, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 462 (913–933) 2006
Analysis of a practical control policy for water storage in two connected dams
Howlett, P; Piantadosi, J; Pearce, Charles, chapter in Continuous optimization: Current trends and modern applications (Springer) 435–450, 2005
Diversity sensitivity and multimodal Bayesian statistical analysis by relative entropy
Leipnik, R; Pearce, Charles, The ANZIAM Journal 47 (277–287) 2005
Elastic plastic analysis of shallow shells - A new approach
Mazumdar, Jagan; Ghosh, Abir; Hewitt, J; Bhattacharya, P, The ANZIAM Journal 47 (121–130) 2005
Examples of unbounded homogeneous domains in complex space
Eastwood, Michael; Isaev, A, Science in China Series A-Mathematics Physics Astronomy 48 (248–261) 2005
Hidden Markov chain filtering for a jump diffusion model
Wu, P; Elliott, Robert, Stochastic Analysis and Applications 23 (153–163) 2005
Hidden Markov filter estimation of the occurrence time of an event in a financial market
Elliott, Robert; Tsoi, A, Stochastic Analysis and Applications 23 (1165–1177) 2005
Meta-analysis of controlled trials of ventilator therapy in acute lung injury and acute respiratory distress syndrome: an alternative perspective
Moran, John; Bersten, A; Solomon, Patricia, Intensive Care Medicine 31 (227–235) 2005
Smoothly parameterized ech cohomology of complex manifolds
Bailey, T; Eastwood, Michael; Gindikin, S, Journal of Geometric Analysis 15 (9–23) 2005
Image processing of finite size rat retinal ganglion cells using multifractal and local connected fractal analysis
Jelinek, H; Cornforth, D; Roberts, Anthony John; Landini, G; Bourke, P; Iorio, A, chapter in AI 2004: Advances in Artificial Intelligence (Springer) 961–966, 2005
On the analysis of a case-control study with differential measurement error
Glonek, Garique, 20th International Workshop on Statistical Modelling, Sydney, Australia 10/07/05
Dixmier traces as singular symmetric functionals and applications to measurable operators
Lord, Steven; Sedaev, A; Sukochev, F, Journal of Functional Analysis 224 (72–106) 2005
Filtering, smoothing and M-ary detection with discrete time poisson observations
Elliott, Robert; Malcolm, William; Aggoun, L, Stochastic Analysis and Applications 23 (939–952) 2005
Finite-dimensional filtering and control for continuous-time nonlinear systems
Elliott, Robert; Aggoun, L; Benmerzouga, A, Stochastic Analysis and Applications 22 (499–505) 2005
Nonlinear analysis of rubber-based polymeric materials with thermal relaxation models
Melnik, R; Strunin, D; Roberts, Anthony John, Numerical Heat Transfer Part A-Applications 47 (549–569) 2005
Smoothly parameterized Cech cohomology of complex manifolds
Bailey, T; Eastwood, Michael; Gindikin, S, Journal of Geometric Analysis 15 (9–23) 2005
A deterministic discretisation-step upper bound for state estimation via Clark transformations
Malcolm, William; Elliott, Robert; Van Der Hoek, John, J.A.M.S.A. Journal of Applied Mathematics and Stochastic Analysis 2004 (371–384) 2004
A sufficient condition for the uniform exponential stability of time-varying systems with noise
Grammel, G; Maizurna, Isna, Nonlinear Analysis-Theory Methods & Applications 56 (951–960) 2004
Gerbes, Clifford Modules and the index theorem
Murray, Michael; Singer, Michael, Annals of Global Analysis and Geometry 26 (355–367) 2004
Reactions to genetically modified food crops and how perception of risks and benefits influences consumers' information gathering
Wilson, Carlene; Evans, G; Leppard, Phillip; Syrette, J, Risk Analysis 24 (1311–1321) 2004
A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogenous anisotropic media
Ang, W; Clements, David; Vahdati, N, Engineering Analysis With Boundary Elements 27 (49–55) 2003
Compact Khler surfaces with trivial canonical bundle
Buchdahl, Nicholas, Annals of Global Analysis and Geometry 23 (189–204) 2003
Complex analysis and the Funk transform
Bailey, T; Eastwood, Michael; Gover, A; Mason, L, Journal of the Korean Mathematical Society 40 (577–593) 2003
Exponential stability and partial averaging
Grammel, G; Maizurna, Isna, Journal of Mathematical Analysis and Applications 283 (276–286) 2003
Hyperbolic monopoles and holomorphic spheres
Murray, Michael; Norbury, Paul; Singer, Michael, Annals of Global Analysis and Geometry 23 (101–128) 2003
Method of best successive approximations for nonlinear operators
Torokhti, Anatoli; Howlett, P; Pearce, Charles, Journal of Computational Analysis and Applications 5 (299–312) 2003
On nonlinear operator approximation with preassigned accuracy
Howlett, P; Pearce, Charles; Torokhti, Anatoli, Journal of Computational Analysis and Applications 5 (273–297) 2003
Rumours, epidemics, and processes of mass action: Synthesis and analysis
Dickinson, Rowland; Pearce, Charles, Mathematical and Computer Modelling 38 (1157–1167) 2003
Resampling-based multiple testing for microarray data analysis (Invited discussion of paper by Ge, Dudoit and Speed)
Glonek, Garique; Solomon, Patricia, Test 12 (50–53) 2003
An analysis of noise enhanced information transmission in an array of comparators
McDonnell, Mark; Abbott, Derek; Pearce, Charles, Microelectronics Journal 33 (1079–1089) 2002
Approximating spectral invariants of Harper operators on graphs
Varghese, Mathai; Yates, Stuart, Journal of Functional Analysis 188 (111–136) 2002
Portfolio optimization, hidden Markov models, and technical analysis of P&F-charts
Elliott, Robert; Hinz, J, International Journal of Theoretical and Applied Finance 5 (385–399) 2002
The Borel-Weil theorem for complex projective space
Eastwood, Michael; Sawon, J, chapter in Invitations to geometry and topology (Oxford University Press) 126–145, 2002
A classification of non-degenerate homogeneous equiaffine hypersurfaces in four complex dimensions
Eastwood, Michael; Ezhov, Vladimir, The Asian Journal of Mathematics 5 (721–740) 2001
An edge-of-the-wedge theorum for hypersurface CR functions
Eastwood, Michael; Graham, C, Journal of Geometric Analysis 11 (589–602) 2001
Csiszr f-divergence, Ostrowski's inequality and mutual information
Dragomir, S; Gluscevic, Vido; Pearce, Charles, Nonlinear Analysis-Theory Methods & Applications 47 (2375–2386) 2001
Equivariant Seiberg-Witten Floer homology
Marcolli, M; Wang, Bai-Ling, Communications in Analysis and Geometry 9 (451–639) 2001
On best-approximation problems for nonlinear operators
Howlett, P; Pearce, Charles; Torokhti, Anatoli, Nonlinear Functional Analysis and Applications 6 (351–368) 2001
On the extended reversed Meir inequality
Guljas, B; Pearce, Charles; Pecaric, Josip, Journal of Computational Analysis and Applications 3 (243–247) 2001
The Mx/G/1 queue with queue length dependent service times
Choi, B; Kim, Y; Shin, Y; Pearce, Charles, J.A.M.S.A. Journal of Applied Mathematics and Stochastic Analysis 14 (399–419) 2001
The modelling and numerical simulation of causal non-linear systems
Howlett, P; Torokhti, Anatoli; Pearce, Charles, Nonlinear Analysis-Theory Methods & Applications 47 (5559–5572) 2001
Complex Quaternionic Kahler Manifolds
Eastwood, Michael, chapter in Further advances in twistor theory. Vol. III, Curved twistor spaces (Chapman & Hall/CRC) 31–34, 2001
Best estimators of second degree for data analysis
Howlett, P; Pearce, Charles; Torokhti, Anatoli, ASMDA 2001, Compiegne, France 12/06/01
A continuous time kronecker's lemma and martingale convergence
Elliott, Robert, Stochastic Analysis and Applications 19 (433–437) 2001
Statistical analysis of medical data: New developments - Book review
Solomon, Patricia, Biometrics 57 (327–328) 2001
Meta-analysis, overviews and publication bias
Solomon, Patricia; Hutton, Jonathon, Statistical Methods in Medical Research 10 (245–250) 2001
A complex from linear elasticity
Eastwood, Michael, 19th Winter School Geometry and Physics, Srni, Czech Republic 09/01/99
Spectral analysis of heart sounds and vibration analysis of heart valves
Mazumdar, Jagan, EMAC 2000, RMIT University, Melbourne, Australia 10/09/00
A martingale analysis of hysteretic overload control
Roughan, Matthew; Pearce, Charles, Advances in Performance Analysis 3 (1–30) 2000
A note on higher cohomology groups of Khler quotients
Wu, Siye, Annals of Global Analysis and Geometry 18 (569–576) 2000
Drawing with complex numbers
Eastwood, Michael; Penrose, R, Mathematical Intelligencer 22 (8–13) 2000
Local Constraints on Einstein-Weyl geometries: The 3-dimensional case
Eastwood, Michael; Tod, K, Annals of Global Analysis and Geometry 18 (1–27) 2000
On Anastassiou's generalizations of the Ostrowski inequality and related results
Pearce, Charles; Pecaric, Josip, Journal of Computational Analysis and Applications 2 (215–276) 2000

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