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April 2018

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Associate Professor Nicholas Buchdahl
Reader in Pure Mathematics

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Professor Finnur Larusson
Associate Professor in Pure Mathematics

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Courses matching "+Complex +analysis"

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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Events matching "+Complex +analysis"

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.
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).
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.
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.
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.
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.
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.
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

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".
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.
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.
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.
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

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.
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.
Turbulent flows, semtex, and rainbows
12:10 Mon 8 Oct, 2012 :: B.21 Ingkarni Wardli :: Ms Sophie Calabretto :: University of Adelaide

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

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).
A stability theorem for elliptic Harnack inequalities
15:10 Fri 5 Apr, 2013 :: B.18 Ingkarni Wardli :: Prof Richard Bass :: University of Connecticut

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.
Holomorphic null curves and the conformal Calabi-Yau problem
12:10 Tue 28 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Franc Forstneric :: University of Ljubljana

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.
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

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.
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

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

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.
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.
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


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.
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

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.
Minimal surfaces and complex analysis
12:10 Fri 24 Mar, 2017 :: Napier 209 :: Antonio Alarcon :: University of Granada

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.
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.
Radial Toeplitz operators on bounded symmetric domains
11:10 Fri 9 Mar, 2018 :: Lower Napier LG11 :: Raul Quiroga-Barranco :: CIMAT, Guanajuato, Mexico

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.

News matching "+Complex +analysis"

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.

Publications matching "+Complex +analysis"

Smoothly parameterized ech cohomology of complex manifolds
Bailey, T; Eastwood, Michael; Gindikin, S, Journal of Geometric Analysis 15 (9–23) 2005
Smoothly parameterized Cech cohomology of complex manifolds
Bailey, T; Eastwood, Michael; Gindikin, S, Journal of Geometric Analysis 15 (9–23) 2005
Complex analysis and the Funk transform
Bailey, T; Eastwood, Michael; Gover, A; Mason, L, Journal of the Korean Mathematical Society 40 (577–593) 2003

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