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Events matching "The two envelope problem"

Making tertiary mathematics more interesting
15:10 Fri 24 Mar, 2006 :: G08 Mathematics Building University of Adelaide :: Prof. Emeritus Neville de Mestre, Faculty of Information Technology, Bond University

For the past few decades, calculus and linear algebra have provided the basis for many university courses in mathematics, science or engineering. However there are other courses, which could be given to motivate the students, particularly those with only a passing love of mathematics. One possible course could show the essential features of how mathematicians solve problems using many different analytical, cerebral and computer skills. In this seminar I will describe such a one-semester course (2 lectures, 2 labs each week), which includes hands-on problem solving and students eventually creating their own problems. One or two exciting problems at first-year level will be developed in detail.
Mathematics of underground mining.
15:10 Fri 12 May, 2006 :: G08 Mathematics Building University of Adelaide :: Prof. Hyam Rubinstein

Underground mining infrastructure involves an interesting range of optimisation problems with geometric constraints. In particular, ramps, drives and tunnels have gradient within a certain prescribed range and turning circles (curvature) are also bounded. Finally obstacles have to be avoided, such as faults, ore bodies themselves and old workings. A group of mathematicians and engineers at Uni of Melb and Uni of SA have been working on this problem for a number of years. I will summarise what we have found and the challenges of working in the mining industry.
Likelihood inference for a problem in particle physics
15:10 Fri 27 Jul, 2007 :: G04 Napier Building University of Adelaide :: Prof. Anthony Davison

The Large Hadron Collider (LHC), a particle accelerator located at CERN, near Geneva, is (currently!) expected to start operation in early 2008. It is located in an underground tunnel 27km in circumference, and when fully operational, will be the world's largest and highest energy particle accelerator. It is hoped that it will provide evidence for the existence of the Higgs boson, the last remaining particle of the so-called Standard Model of particle physics. The quantity of data that will be generated by the LHC is roughly equivalent to that of the European telecommunications network, but this will be boiled down to just a few numbers. After a brief introduction, this talk will outline elements of the statistical problem of detecting the presence of a particle, and then sketch how higher order likelihood asymptotics may be used for signal detection in this context. The work is joint with Nicola Sartori, of the Università Ca' Foscari, in Venice.
Likelihood inference for a problem in particle physics MATT
15:10 Fri 27 Jul, 2007 :: G04 Napier Building University of Adelaide :: Prof. Anthony Davison

Riemann's Hypothesis
15:10 Fri 31 Aug, 2007 :: G08 Mathematics building University of Adelaide :: Emeritus Prof. E. O. Tuck

Riemann's hypothesis (that all non-trivial zeros of the zeta function have real part one-half) is the most famous currently unproved conjecture in mathematics, and a \\$1M prize awaits its proof. The mathematical statement of this problem is only at about second-year undergraduate level; after all, the zeta function is much like the trigonometric sine function, and all (?) second-year students know that all zeros of the sine function are (real) integer multiples of $\\pi$. Many of the steps apparently needed to make progress on the proof are also not much more complicated than that level. Some of these elementary steps, together with numerical explorations, will be described here. Nevertheless the Riemann hypothesis has defied proof so far, and very complicated and advanced abstract mathematics (that will NOT be described here) is often brought to bear on it. Does it need abstract mathematics, or just a flash of elementary inspiration?
Groundwater: using mathematics to solve our water crisis
13:10 Wed 9 Apr, 2008 :: Napier 210 :: Dr Michael Teubner

'The driest state in the driest continent' is how South Australia used to be described. And that was before the drought! Now we have severe water restrictions, dead lawns, and dying gardens. But this need not be the case. Mathematics to the rescue! Groundwater exists below much of the Adelaide metro area. We can store winter stormwater in the ground and use it when we need it in summer. But we need mathematical models to understand where groundwater exists, where we can inject stormwater and how much can be stored, and where we can extract it: all through mathematical models. Come along and see that we don't have a water problem, we have a water management problem.
Puzzle-based learning: Introduction to mathematics
15:10 Fri 23 May, 2008 :: LG29 Napier Building University of Adelaide :: Prof. Zbigniew Michalewicz :: School of Computer Science, University of Adelaide

Media...
The talk addresses a gap in the educational curriculum for 1st year students by proposing a new course that aims at getting students to think about how to frame and solve unstructured problems. The idea is to increase the student's mathematical awareness and problem-solving skills by discussing a variety of puzzles. The talk makes an argument that this approach - called Puzzle-Based Learning - is very beneficial for introducing mathematics, critical thinking, and problem-solving skills.

The new course has been approved by the University of Adelaide for Faculty of Engineering, Computer Science, and Mathematics. Many other universities are in the process of introducing such a course. The course will be offered in two versions: (a) full-semester course and (b) a unit within general course (e.g. Introduction to Engineering). All teaching materials (power point slides, assignments, etc.) are being prepared. The new textbook (Puzzle-Based Learning: Introduction to Critical Thinking, Mathematics, and Problem Solving) will be available from June 2008. The talk provides additional information on this development.

For further information see http://www.PuzzleBasedlearning.edu.au/

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.
Elliptic equation for diffusion-advection flows
15:10 Fri 15 Aug, 2008 :: G03 Napier Building University of Adelaide :: Prof. Pavel Bedrikovsetsky :: Australian School of Petroleum Science, University of Adelaide.

The standard diffusion equation is obtained by Einstein's method and its generalisation, Fokker-Plank-Kolmogorov-Feller theory. The time between jumps in Einstein derivation is constant.

We discuss random walks with residence time distribution, which occurs for flows of solutes and suspensions/colloids in porous media, CO2 sequestration in coal mines, several processes in chemical, petroleum and environmental engineering. The rigorous application of the Einstein's method results in new equation, containing the time and the mixed dispersion terms expressing the dispersion of the particle time steps.

Usually, adding the second time derivative results in additional initial data. For the equation derived, the condition of limited solution when time tends to infinity provides with uniqueness of the Caushy problem solution.

The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions of the direct simulation.

Key Predistribution in Grid-Based Wireless Sensor Networks
15:10 Fri 12 Dec, 2008 :: Napier G03 :: Dr Maura Paterson :: Information Security Group at Royal Holloway, University of London.

Wireless sensors are small, battery-powered devices that are deployed to measure quantities such as temperature within a given region, then form a wireless network to transmit and process the data they collect. We discuss the problem of distributing symmetric cryptographic keys to the nodes of a wireless sensor network in the case where the sensors are arranged in a square or hexagonal grid, and we propose a key predistribution scheme for such networks that is based on Costas arrays. We introduce more general structures known as distinct-difference configurations, and show that they provide a flexible choice of parameters in our scheme, leading to more efficient performance than that achieved by prior schemes from the literature.
Hunting Non-linear Mathematical Butterflies
15:10 Fri 23 Jan, 2009 :: Napier LG29 :: Prof Nalini Joshi :: University of Sydney

The utility of mathematical models relies on their ability to predict the future from a known set of initial states. But there are non-linear systems, like the weather, where future behaviours are unpredictable unless their initial state is known to infinite precision. This is the butterfly effect. I will show how to analyse functions to overcome this problem for the classical Painleve equations, differential equations that provide archetypical non-linear models of modern physics.
Sloshing in tanks of liquefied natural gas (LNG) vessels
15:10 Wed 22 Apr, 2009 :: Napier LG29 :: Prof. Frederic Dias :: ENS, Cachan

The last scientific conversation I had with Ernie Tuck was on liquid impact. As a matter of fact, we discussed the paper by J.H. Milgram, Journal of Fluid Mechanics 37 (1969), entitled "The motion of a fluid in a cylindrical container with a free surface following vertical impact." Liquid impact is a key issue in sloshing and in particular in sloshing in tanks of LNG vessels. Numerical simulations of sloshing have been performed by various groups, using various types of numerical methods. In terms of the numerical results, the outcome is often impressive, but the question remains of how relevant these results are when it comes to determining impact pressures. The numerical models are too simplified to reproduce the high variability of the measured pressures. In fact, for the time being, it is not possible to simulate accurately both global and local effects. Unfortunately it appears that local effects predominate over global effects when the behaviour of pressures is considered. Having said this, it is important to point out that numerical studies can be quite useful to perform sensitivity analyses in idealized conditions such as a liquid mass falling under gravity on top of a horizontal wall and then spreading along the lateral sides. Simple analytical models inspired by numerical results on idealized problems can also be useful to predict trends. The talk is organized as follows: After a brief introduction on the sloshing problem and on scaling laws, it will be explained to what extent numerical studies can be used to improve our understanding of impact pressures. Results on a liquid mass hitting a wall obtained by a finite-volume code with interface reconstruction as well as results obtained by a simple analytical model will be shown to reproduce the trends of experiments on sloshing. This is joint work with L. Brosset (GazTransport & Technigaz), J.-M. Ghidaglia (ENS Cachan) and J.-P. Braeunig (INRIA).
Quadrature domains, p-Laplacian growth, and bubbles contracting in Hele-Shaw cells with a power-law fluid.
15:10 Mon 15 Jun, 2009 :: Napier LG24 :: Dr Scott McCue :: Queensland University Technology

The classical Hele-Shaw flow problem is related to Laplacian growth and null-quadrature domains. A generalisation is constructed for power-law fluids, governed by the p-Laplace equation, and a number of results are established that are analogous to the classical case. Both fluid clearance and bubble extinction is considered, and by considering two extremes of extinction behaviour, a rather complete asymptotic description of possible behaviours is found.
Nonlinear diffusion-driven flow in a stratified viscous fluid
15:00 Fri 26 Jun, 2009 :: Macbeth Lecture Theatre :: Associate Prof Michael Page :: Monash University

In 1970, two independent studies (by Wunsch and Phillips) of the behaviour of a linear density-stratified viscous fluid in a closed container demonstrated a slow flow can be generated simply due to the container having a sloping boundary surface This remarkable motion is generated as a result of the curvature of the lines of constant density near any sloping surface, which in turn enables a zero normal-flux condition on the density to be satisfied along that boundary. When the Rayleigh number is large (or equivalently Wunsch's parameter $R$ is small) this motion is concentrated in the near vicinity of the sloping surface, in a thin `buoyancy layer' that has many similarities to an Ekman layer in a rotating fluid.

A number of studies have since considered the consequences of this type of `diffusively-driven' flow in a semi-infinite domain, including in the deep ocean and with turbulent effects included. More recently, Page & Johnson (2008) described a steady linear theory for the broader-scale mass recirculation in a closed container and demonstrated that, unlike in previous studies, it is possible for the buoyancy layer to entrain fluid from that recirculation. That work has since been extended (Page & Johnson, 2009) to the nonlinear regime of the problem and some of the similarities to and differences from the linear case will be described in this talk. Simple and elegant analytical solutions in the limit as $R \to 0$ still exist in some situations, and they will be compared with numerical simulations in a tilted square container at small values of $R$. Further work on both the unsteady flow properties and the flow for other geometrical configurations will also be described.

Unsolvable problems in mathematics
15:10 Fri 3 Jul, 2009 :: Badger Labs G13 Macbeth Lecture Theatre :: Prof Greg Hjorth :: University of Melbourne

In the 1900 International Congress of Mathematicians David Hilbert proposed a list of 23 landmark mathematical problems. The first of these problems was shown by Paul Cohen in 1963 to be undecidable—which is to say, in informal language, that it was in principle completely unsolvable. The tenth problem was shown by Yuri Matiyasevich to be unsolvable in 1970. These developments would very likely have been profoundly surprising, perhaps even disturbing, to Hilbert. I want to review some of the general history of unsolvable problems. As much as reasonably possible in the time allowed, I hope to give the audience a sense of why the appearance of unsolvable problems in mathematics was inevitable, and perhaps even desirable.
Curved pipe flow and its stability
15:10 Fri 11 Sep, 2009 :: Badger labs G13 Macbeth Lecture Theatre :: Dr Richard Clarke :: University of Auckland

The unsteady flow of a viscous fluid through a curved pipe is a widely occuring and well studied problem. The stability of such flows, however, has largely been overlooked; this is in marked contrast to flow through a straight-pipe, examination of which forms a cornerstone of hydrodynamic stability theory. Importantly, however, flow through a curved pipe exhibits an array of flow structures that are simply not present in the zero curvature limit, and it is natural to expect these to substantially impact upon the flow's stability. By considering two very different kinds of flows through a curved pipe, we illustrate that this can indeed be the case.
Stable commutator length
13:40 Fri 25 Sep, 2009 :: Napier 102 :: Prof Danny Calegari :: California Institute of Technology

Stable commutator length answers the question: "what is the simplest surface in a given space with prescribed boundary?" where "simplest" is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). In these talks we will discuss how scl in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi- dimensional continued fractions and Klein polyhedra. Danny Calegari is the Richard Merkin Professor of Mathematics at the California Institute of Technology, and is one of the recipients of the 2009 Clay Research Award for his work in geometric topology and geometric group theory. He received a B.A. in 1994 from the University of Melbourne, and a Ph.D. in 2000 from the University of California, Berkeley under the joint supervision of Andrew Casson and William Thurston. From 2000 to 2002 he was Benjamin Peirce Assistant Professor at Harvard University, after which he joined the Caltech faculty; he became Richard Merkin Professor in 2007.
Contemporary frontiers in statistics
15:10 Mon 28 Sep, 2009 :: Badger Labs G31 Macbeth Lectrue :: Prof. Peter Hall :: University of Melbourne

The availability of powerful computing equipment has had a dramatic impact on statistical methods and thinking, changing forever the way data are analysed. New data types, larger quantities of data, and new classes of research problem are all motivating new statistical methods. We shall give examples of each of these issues, and discuss the current and future directions of frontier problems in statistics.
Is the price really right?
12:10 Thu 22 Oct, 2009 :: Napier 210 :: Mr Sam Cohen :: University of Adelaide

Media...
Making decisions when outcomes are uncertain is a common problem we all face. In this talk I will outline some recent developments on this question from the mathematics of finance-the theory of risk measures and nonlinear expectations. I will also talk about how decisions are currently made in the finance industry, and how some simple mathematics can show where these systems are open to abuse.
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.
Integrable systems: noncommutative versus commutative
14:10 Thu 4 Mar, 2010 :: School Board Room :: Dr Cornelia Schiebold :: Mid Sweden University

After a general introduction to integrable systems, we will explain an approach to their solution theory, which is based on Banach space theory. The main point is first to shift attention to noncommutative integrable systems and then to extract information about the original setting via projection techniques. The resulting solution formulas turn out to be particularly well-suited to the qualitative study of certain solution classes. We will show how one can obtain a complete asymptotic description of the so called multiple pole solutions, a problem that was only treated for special cases before.
The fluid mechanics of gels used in tissue engineering
15:10 Fri 9 Apr, 2010 :: Santos Lecture Theatre :: Dr Edward Green :: University of Western Australia

Tissue engineering could be called 'the science of spare parts'. Although currently in its infancy, its long-term aim is to grow functional tissues and organs in vitro to replace those which have become defective through age, trauma or disease. Recent experiments have shown that mechanical interactions between cells and the materials in which they are grown have an important influence on tissue architecture, but in order to understand these effects, we first need to understand the mechanics of the gels themselves.

Many biological gels (e.g. collagen) used in tissue engineering have a fibrous microstructure which affects the way forces are transmitted through the material, and which in turn affects cell migration and other behaviours. I will present a simple continuum model of gel mechanics, based on treating the gel as a transversely isotropic viscous material. Two canonical problems are considered involving thin two-dimensional films: extensional flow, and squeezing flow of the fluid between two rigid plates. Neglecting inertia, gravity and surface tension, in each regime we can exploit the thin geometry to obtain a leading-order problem which is sufficiently tractable to allow the use of analytical methods. I discuss how these results could be exploited practically to determine the mechanical properties of real gels. If time permits, I will also talk about work currently in progress which explores the interaction between gel mechanics and cell behaviour.

Two problems in porous media flow
15:10 Tue 11 May, 2010 :: Santos Lecture Theatre :: A/Prof Graeme Hocking :: Murdoch University

I will discuss two problems in porous media flow.

On a tropical island, fresh water may sit in the soil beneath the ground, floating on the ocean's salt water. This water is a valuable resource for the inhabitants, but requires sufficient rainfall to recharge the lens. In this paper, Green's functions are used to derive an integral equation to satisfy all of the conditions except those on the interfaces, which are then solved for numerically. Conditions under which the lens can be maintained will be described. This is work I did with an Honours student, Sue Chen, who is now at U. Melbourne.

In the second problem, I will discuss an "exact" solution to a problem in withdrawal from an unconfined aquifer. The problem formulation gives rise to a singular integral equation that can be solved using a nice orthogonality result I first met in airfoil theory. This is work with Hong Zhang from Griffith University.

A variance constraining ensemble Kalman filter: how to improve forecast using climatic data of unobserved variables
15:10 Fri 28 May, 2010 :: Santos Lecture Theatre :: A/Prof Georg Gottwald :: The University of Sydney

Data assimilation aims to solve one of the fundamental problems ofnumerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system. A standard tool in attacking this filtering problem is the Kalman filter.

We consider the problem when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. We derive the corresponding Kalman filter in a variational setting.

We analyze the variance constraining Kalman filter (VCKF) filter for a simple linear toy model and determine its range of optimal performance. We explore the variance constraining Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information on the variance on some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.

Using methods from dynamical systems theory we then systems where the un-observed variables evolve deterministically but chaotically on a fast time scale.

This is joint work with Lewis Mitchell and Sebastian Reich.

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.
Counting lattice points in polytopes and geometry
15:10 Fri 6 Aug, 2010 :: Napier G04 :: Dr Paul Norbury :: University of Melbourne

Counting lattice points in polytopes arises in many areas of pure and applied mathematics. A basic counting problem is this: how many different ways can one give change of 1 dollar into 5,10, 20 and 50 cent coins? This problem counts lattice points in a tetrahedron, and if there also must be exactly 10 coins then it counts lattice points in a triangle. The number of lattice points in polytopes can be used to measure the robustness of a computer network, or in statistics to test independence of characteristics of samples. I will describe the general structure of lattice point counts and the difficulty of calculations. I will then describe a particular lattice point count in which the structure simplifies considerably allowing one to calculate easily. I will spend a brief time at the end describing how this is related to the moduli space of Riemann surfaces.
The two envelope problem
12:10 Wed 11 Aug, 2010 :: Napier 210 :: A/Prof Gary Glonek :: University of Adelaide

Media...
The two envelope problem is a long standing paradox in probability theory. Although its formulation has elements in common with the celebrated Monty Hall problem, the underlying paradox is apparently far more subtle. In this talk, the problem will be explained and various aspects of the paradox will be discussed. Connections to Bayesian inference and other areas of statistics will be explored.
Totally disconnected, locally compact groups
15:10 Fri 17 Sep, 2010 :: Napier G04 :: Prof George Willis :: University of Newcastle

Locally compact groups occur in many branches of mathematics. Their study falls into two cases: connected groups, which occur as automorphisms of smooth structures such as spheres for example; and totally disconnected groups, which occur as automorphisms of discrete structures such as trees. The talk will give an overview of the currently developing structure theory of totally disconnected locally compact groups. Techniques for analysing totally disconnected groups will be described that correspond to the familiar Lie group methods used to treat connected groups. These techniques played an essential role in the recent solution of a problem raised by R. Zimmer and G. Margulis concerning commensurated subgroups of arithmetic groups.
Hugs not drugs
15:10 Mon 20 Sep, 2010 :: Ingkarni Wardli B17 :: Dr Scott McCue :: Queensland University of Technology

I will discuss a model for drug diffusion that involves a Stefan problem with a "kinetic undercooling". I like Stefan problems, so I like this model. I like drugs too, but only legal ones of course. Anyway, it turns out that in some parameter regimes, this sophisticated moving boundary problem hardly works better than a simple linear undergraduate model (there's a lesson here for mathematical modelling). On the other hand, for certain polymer capsules, the results are interesting and suggest new means for controlled drug delivery. If time permits, I may discuss certain asymptotic limits that are of interest from a Stefan problem perspective. Finally, I won't bring any drugs with me to the seminar, but I'm willing to provide hugs if necessary.
IGA-AMSI Workshop: Dirac operators in geometry, topology, representation theory, and physics
10:00 Mon 18 Oct, 2010 :: 7.15 Ingkarni Wardli :: Prof Dan Freed :: University of Texas, Austin

Lecture Series by Dan Freed (University of Texas, Austin). Dirac introduced his eponymous operator to describe electrons in quantum theory. It was rediscovered by Atiyah and Singer in their study of the index problem on manifolds. In these lectures we explore new theorems and applications. Several of these also involve K-theory in its recent twisted and differential variations. These lectures will be supplemented by additional talks by invited speakers. For more details, please see the conference webpage: http://www.iga.adelaide.edu.au/workshops/WorkshopOct2010/
Slippery issues in nano- and microscale fluid flows
11:10 Tue 30 Nov, 2010 :: Innova teaching suite B21 :: Dr Shaun C. Hendy :: Victoria University of Wellington

The no-slip boundary condition was considered to have been experimentally established for the flow of simple liquids over solid surfaces in the early 20th century. Nonetheless the refinement of a number of measurement techniques has recently led to the observation of nano- and microscale violations of the no-slip boundary condition by simple fluids flowing over non-wetting surfaces. However it is important to distinguish between intrinsic slip, which arises solely from the chemical interaction between the liquid and a homogeneous, atomically flat surface and effective slip, typically measured in macroscopic experiments, which emerges from the interaction of microscopic chemical heterogeneity, roughness and contaminants. Here we consider the role of both intrinsic and effective slip boundary conditions in nanoscale and microscale fluid flows using a theoretical approach, complemented by molecular dynamics simulations, and experimental evidence where available. Firstly, we consider nanoscale flows in small capillaries, including carbon nanotubes, where we have developed and solved a generalised Lucas-Washburn equation that incorporates slip to describe the uptake of droplets. We then consider the general problem of relating effective slip to microscopic intrinsic slip and roughness, and discuss several cases where we have been able to solve this problem analytically. Finally, we look at applications of these results to carbon nanotube growth, self-cleaning surfaces, catalysis, and putting insulation in your roof.
How round is your triangle, square, pentagon, ...?
12:10 Wed 6 Apr, 2011 :: Napier 210 :: Dr Barry Cox :: University of Adelaide

Media...
Most of us are familiar with the problem of making circular holes in wood or other material. For smaller diameter holes we typically use a drill, and for larger diameter holes a spade-bit, hole-saw or plunge router may be used. However for some applications, like mortise-and-tenon joints, what is needed is a tool that will produce a hole with a cross-section that is something other than a circle. In this talk we look at curves that may be used as the basis for a device that will produce holes with a cross-section of an equilateral triangle, square, or any regular polygon. Along the way we will touch on areas of engineering, algebra, geometry, calculus, Gothic art and architecture.
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.
The Cauchy integral formula
12:10 Mon 9 May, 2011 :: 5.57 Ingkarni Wardli :: Stephen Wade :: University of Adelaide

In this talk I will explain a simple method used for calculating the Hilbert transform of an analytic function, and provide some assurance that this isn't a bad thing to do in spite of the somewhat ominous presence of infinite areas. As it turns out this type of integral is not without an application, as will be demonstrated by one application to a problem in fluid mechanics.
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.
The Extended-Domain-Eigenfunction Method: making old mathematics work for new problems
15:10 Fri 13 May, 2011 :: 7.15 Ingkarni Wardli :: Prof Stan Miklavcic :: University of South Australia

Media...
Standard analytical solutions to elliptic boundary value problems on asymmetric domains are rarely, if ever, obtainable. Several years ago I proposed a solution technique to cope with such complicated domains. It involves the embedding of the original domain into one with simple boundaries where the classical eigenfunction solution approach can be used. The solution in the larger domain, when restricted to the original domain is then the solution of the original boundary value problem. In this talk I will present supporting theory for this idea, some numerical results for the particular case of the Laplace equation and the Stokes flow equations in two-dimensions and discuss advantages and limitations of the proposal.
Statistical modelling in economic forecasting: semi-parametrically spatio-temporal approach
12:10 Mon 23 May, 2011 :: 5.57 Ingkarni Wardli :: Dawlah Alsulami :: University of Adelaide

How to model spatio-temporal variation of housing prices is an important and challenging problem as it is of vital importance for both investors and policy makersto assess any movement in housing prices. In this seminar I will talk about the proposed model to estimate any movement in housing prices and measure the risk more accurately.
Inference and optimal design for percolation and general random graph models (Part I)
09:30 Wed 8 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Andrei Bejan :: The University of Cambridge

The problem of optimal arrangement of nodes of a random weighted graph is discussed in this workshop. The nodes of graphs under study are fixed, but their edges are random and established according to the so called edge-probability function. This function is assumed to depend on the weights attributed to the pairs of graph nodes (or distances between them) and a statistical parameter. It is the purpose of experimentation to make inference on the statistical parameter and thus to extract as much information about it as possible. We also distinguish between two different experimentation scenarios: progressive and instructive designs.

We adopt a utility-based Bayesian framework to tackle the optimal design problem for random graphs of this kind. Simulation based optimisation methods, mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain the solution. We study optimal design problem for the inference based on partial observations of random graphs by employing data augmentation technique. We prove that the infinitely growing or diminishing node configurations asymptotically represent the worst node arrangements. We also obtain the exact solution to the optimal design problem for proximity (geometric) graphs and numerical solution for graphs with threshold edge-probability functions.

We consider inference and optimal design problems for finite clusters from bond percolation on the integer lattice $\mathbb{Z}^d$ and derive a range of both numerical and analytical results for these graphs. We introduce inner-outer plots by deleting some of the lattice nodes and show that the ëmostly populatedí designs are not necessarily optimal in the case of incomplete observations under both progressive and instructive design scenarios. Some of the obtained results may generalise to other lattices.

Inference and optimal design for percolation and general random graph models (Part II)
10:50 Wed 8 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Andrei Bejan :: The University of Cambridge

The problem of optimal arrangement of nodes of a random weighted graph is discussed in this workshop. The nodes of graphs under study are fixed, but their edges are random and established according to the so called edge-probability function. This function is assumed to depend on the weights attributed to the pairs of graph nodes (or distances between them) and a statistical parameter. It is the purpose of experimentation to make inference on the statistical parameter and thus to extract as much information about it as possible. We also distinguish between two different experimentation scenarios: progressive and instructive designs.

We adopt a utility-based Bayesian framework to tackle the optimal design problem for random graphs of this kind. Simulation based optimisation methods, mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain the solution. We study optimal design problem for the inference based on partial observations of random graphs by employing data augmentation technique. We prove that the infinitely growing or diminishing node configurations asymptotically represent the worst node arrangements. We also obtain the exact solution to the optimal design problem for proximity (geometric) graphs and numerical solution for graphs with threshold edge-probability functions.

We consider inference and optimal design problems for finite clusters from bond percolation on the integer lattice $\mathbb{Z}^d$ and derive a range of both numerical and analytical results for these graphs. We introduce inner-outer plots by deleting some of the lattice nodes and show that the ëmostly populatedí designs are not necessarily optimal in the case of incomplete observations under both progressive and instructive design scenarios. Some of the obtained results may generalise to other lattices.

Stochastic models of reaction diffusion
15:10 Fri 17 Jun, 2011 :: 7.15 Ingkarni Wardli :: Prof Jon Chapman :: Oxford University

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We consider two different position jump processes: (i) a random walk on a lattice (ii) the Euler scheme for the Smoluchowski differential equation. Both of these reduce to the diffusion equation as the time step and size of the jump tend to zero. We consider the problem of adding chemical reactions to these processes, both at a surface and in the bulk. We show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which stochastic model for diffusion is used.
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.
Towards Rogers-Ramanujan identities for the Lie algebra A_n
13:10 Fri 5 Aug, 2011 :: B.19 Ingkarni Wardli :: Prof Ole Warnaar :: University of Queensland

The Rogers-Ramanujan identities are a pair of q-series identities proved by Leonard Rogers in 1894 which became famous two decades later as conjectures of Srinivasa Ramanujan. Since the 1980s it is known that the Rogers-Ramanujan identities are in fact identities for characters of certain modules for the affine Lie algebra A_1. This poses the obvious question as to whether there exist Rogers-Ramanujan identities for higher rank affine Lie algebras. In this talk I will describe some recent progress on this problem. I will also discuss a seemingly mysterious connection with the representation theory of quivers over finite fields.
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.
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.
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.
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.
Fluid mechanics: what's maths got to do with it?
13:10 Tue 20 Mar, 2012 :: 7.15 Ingkarni Wardli :: A/Prof Jim Denier :: School of Mathematical Sciences

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We've all heard about the grand challenges in mathematics. There was the Poincare Conjecture, which has now been resolved. There is the Riemann Hypothesis which many are seeking to prove. But one of the most intriguing is the so called "Navier-Stokes Equations" problem, intriguing because it not only involves some wickedly difficult mathematics but also involves questions about our deep understanding of nature as encountered in the flow of fluids. This talk will introduce the problem (without the wickedly difficult mathematics) and discuss some of the consequences of its resolution.
Bundle gerbes and the Faddeev-Mickelsson-Shatashvili anomaly
13:10 Fri 30 Mar, 2012 :: B.20 Ingkarni Wardli :: Dr Raymond Vozzo :: University of Adelaide

The Faddeev-Mickelsson-Shatashvili anomaly arises in the quantisation of fermions interacting with external gauge potentials. Mathematically, it can be described as a certain lifting problem for an extension of groups. The theory of bundle gerbes is very useful for studying lifting problems, however it only applies in the case of a central extension whereas in the study of the FMS anomaly the relevant extension is non-central. In this talk I will explain how to describe this anomaly indirectly using bundle gerbes and how to use a generalisation of bundle gerbes to describe the (non-central) lifting problem directly. This is joint work with Pedram Hekmati, Michael Murray and Danny Stevenson.
The Kazdan-Warner equation
12:10 Mon 2 Apr, 2012 :: 5.57 Ingkarni Wardli :: Mr Damien Warman :: University of Adelaide

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We look at an equation arising from the differential-geometric problem of specifying the scalar curvature of a manifold.
A Problem of Siegel
13:10 Fri 27 Apr, 2012 :: B.20 Ingkarni Wardli :: Dr Brent Everitt :: University of York

The first explicit examples of orientable hyperbolic 3-manifolds were constructed by Weber, Siefert, and Lobell in the early 1930's. In the subsequent decades the world of hyperbolic n-manifolds has grown into an extraordinarily rich one. Its sociology is best understood through the eyes of invariants, and for hyperbolic manifolds the most important invariant is volume. Viewed this way the n-dimensional hyperbolic manifolds, for fixed n, look like a well-ordered subset of the reals (a discrete set even, when n is not 3). So we are naturally led to the (manifold) Siegel problem: for a given n, determine the minimum possible volume obtained by an orientable hyperbolic n-manifold. It is a problem with a long and venerable history. In this talk I will describe a unified solution to the problem in low even dimensions, one of which at least is new. Joint work with John Ratcliffe and Steve Tschantz (Vanderbilt).
Modelling protective anti-tumour immunity using a hybrid agent-based and delay differential equation approach
15:10 Fri 11 May, 2012 :: B.21 Ingkarni Wardli :: Dr Peter Kim :: University of Sydney

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Although cancers seem to consistently evade current medical treatments, the body's immune defences seem quite effective at controlling incipient tumours. Understanding how our immune systems provide such protection against early-stage tumours and how this protection could be lost will provide insight into designing next-generation immune therapies against cancer. To engage this problem, we formulate a mathematical model of the immune response against small, incipient tumours. The model considers the initial stimulation of the immune response in lymph nodes and the resulting immune attack on the tumour and is formulated as a hybrid agent-based and delay differential equation model.
Computational complexity, taut structures and triangulations
13:10 Fri 18 May, 2012 :: Napier LG28 :: Dr Benjamin Burton :: University of Queensland

There are many interesting and difficult algorithmic problems in low-dimensional topology. Here we study the problem of finding a taut structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut structures is "hard", in the sense that it is NP-complete. We also prove that detecting taut structures is "not too hard", by showing it to be fixed-parameter tractable. This is joint work with Jonathan Spreer.
P or NP: this is the question
13:10 Tue 22 May, 2012 :: 7.15 Ingkarni Wardli :: Dr Ali Eshragh :: School of Mathematical Sciences

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Up to early 70's, the main concentration of mathematicians was the design of algorithms. However, the advent of computers changed this focus from not just the design of an algorithm but also to the most efficient algorithm. This created a new field of research, namely the complexity of algorithms, and the associated problem "Is P equal to NP?" was born. The latter question has been unknown for more than four decades and is one of the most famous open problems of the 21st century. Any person who can solve this problem will be awarded US$1,000,000 by the Clay Institute. In this talk, we are going to introduce this problem through simple examples and explain one of the intriguing approaches that may help to solve it.
A brief introduction to Support Vector Machines
12:30 Mon 4 Jun, 2012 :: 5.57 Ingkarni Wardli :: Mr Tyman Stanford :: University of Adelaide

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Support Vector Machines (SVMs) are used in a variety of contexts for a range of purposes including regression, feature selection and classification. To convey the basic principles of SVMs, this presentation will focus on the application of SVMs to classification. Classification (or discrimination), in a statistical sense, is supervised model creation for the purpose of assigning future observations to a group or class. An example might be determining healthy or diseased labels to patients from p characteristics obtained from a blood sample. While SVMs are widely used, they are most successful when the data have one or more of the following properties: The data are not consistent with a standard probability distribution. The number of observations, n, used to create the model is less than the number of predictive features, p. (The so-called small-n, big-p problem.) The decision boundary between the classes is likely to be non-linear in the feature space. I will present a short overview of how SVMs are constructed, keeping in mind their purpose. As this presentation is part of a double post-grad seminar, I will keep it to a maximum of 15 minutes.
Notions of non-commutative metric spaces; why and how
15:10 Fri 15 Jun, 2012 :: B.21 Ingkarni Wardli :: Dr Ittay Weiss :: The University of the South Pacific

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The classical notion of metric space includes the axiom of symmetry: d(x,y)=d(y,x). Some applications of metric techniques to problems in computer graphics, concurrency, and physics (to mention a few) are seriously stressing the limitations imposed by symmetry, resulting in various relaxations of it. I will review some of the motivating problems that seem to require non-symmetry and then review some of the suggested models to deal with the problem. My review will be critical to the topological implications (which are often unpleasant) of some of the models and I will present metric 1-spaces, a new notion of generalized metric spaces.
The Four Colour Theorem
11:10 Mon 23 Jul, 2012 :: B.17 Ingkarni Wardli :: Mr Vincent Schlegel :: University of Adelaide

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Arguably the most famous problem in discrete mathematics, the Four Colour Theorem was first conjectured in 1852 by South African mathematician Francis Guthrie. For 124 years, it defied many attempts to prove and disprove it. I will talk briefly about some of the rich history of this result, including some of the graph-theoretic techniques used in the 1976 Appel-Haken proof.
The fundamental theorems of invariant theory, classical and quantum
15:10 Fri 10 Aug, 2012 :: B.21 Ingkarni Wardli :: Prof Gus Lehrer :: The University of Sydney

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Let V = C^n, and let (-,-) be a non-degenerate bilinear form on V , which is either symmetric or anti-symmetric. Write G for the isometry group of (V , (-,-)); thus G = O_n (C) or Sp_n (C). The first fundamental theorem (FFT) provides a set of generators for End_G(V^{\otimes r} ) (r = 1, 2, . . . ), while the second fundamental theorem (SFT) gives all relations among the generators. In 1937, Brauer formulated the FFT in terms of his celebrated 'Brauer algebra' B_r (\pm n), but there has hitherto been no similar version of the SFT. One problem has been the generic non-semisimplicity of B_r (\pm n), which caused H Weyl to call it, in his work on invariants 'that enigmatic algebra'. I shall present a solution to this problem, which shows that there is a single idempotent in B_r (\pm n), which describes all the relations. The proof is through a new 'Brauer category', in which the fundamental theorems are easily formulated, and where a calculus of tangles may be used to prove these results. There are quantum analogues of the fundamental theorems which I shall also discuss. There are numerous applications in representation theory, geometry and topology. This is joint work with Ruibin Zhang.
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.
Continuous random walk models for solute transport in porous media
15:10 Fri 17 Aug, 2012 :: B.21 Ingkarni Wardli :: Prof Pavel Bedrikovetski :: The University of Adelaide

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The classical diffusion (thermal conductivity) equation was derived from the Master random walk equation and is parabolic. The main assumption was a probabilistic distribution of the jump length while the jump time is constant. Distribution of the jump time along with the jump length adds the second time derivative into the averaged equations, but the equation becomes ... elliptic! Where from to take an extra initial condition? We discuss how to pose the well-posed flow problem, exact 1d solution and numerous engineering applications. This is joint work with A. Shapiro and H. Yuan.
Wave propagation in disordered media
15:10 Fri 31 Aug, 2012 :: B.21 Ingkarni Wardli :: Dr Luke Bennetts :: The University of Adelaide

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Problems involving wave propagation through systems composed of arrays of scattering sources embedded in some background medium will be considered. For example, in a fluids setting, the background medium is the open ocean surface and the scatterers are floating bodies, such as wave energy devices. Waves propagate in very different ways if the system is structured or disordered. If the disorder is random the problem is to determine the `effective' wave propagation properties by considering the ensemble average over all possible realisations of the system. I will talk about semi-analytical (i.e. low numerical cost) approaches to determining the effective properties.
The Wonderful World of Interval Arithmetic
12:30 Mon 10 Sep, 2012 :: B.21 Ingkarni Wardli :: Ms Mingmei Teo :: University of Adelaide

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There are many situations where we round off answers or give approximations to solutions to equations. Are we happy to do so or are there ways we can overcome this problem? What about providing intervals in which the true solution lies? An example of this is when Archimedes was able to contain \pi by taking a circle between inscribed and circumscribed polygons and take an increasing number of sides of the polygons. In this talk, I will explain a variety of things to do with interval arithmetic. These range from why interval arithmetic is useful to us, some basics of interval arithmetic and also some interesting and cool properties of intervals. I will also discuss briefly how I use it in my project.
The advection-diffusion-reaction equation on the surface of the sphere
12:10 Mon 24 Sep, 2012 :: B.21 Ingkarni Wardli :: Mr Kale Davies :: University of Adelaide

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We aim to solve the advection-diffusion-reaction equation on the surface of a sphere. In order to do this we will be required to utilise spherical harmonics, a set of solutions to Laplace's equation in spherical coordinates. Upon solving the equations, we aim to find a set of parameters that cause a localised concentration to be maintained in the flow, referred to as a hotspot. In this talk I will discuss the techniques that are required to numerically solve this problem and the issues that occur/how to deal with these issues when searching for hotspot solutions.
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.
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.
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.
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.
Crystallographic groups I: the classical theory
12:10 Fri 17 May, 2013 :: Ingkarni Wardli B19 :: Dr Wolfgang Globke :: University of Adelaide

A discrete isometry group acting properly discontinuously on the n-dimensional Euclidean space with compact quotient is called a crystallographic group. This name reflects the fact that in dimension n=3 their compact fundamental domains resemble a space-filling crystal pattern. For higher dimensions, Hilbert posed his famous 18th problem: "Is there in n-dimensional Euclidean space only a finite number of essentially different kinds of groups of motions with a [compact] fundamental region?" This problem was solved by Bieberbach when he proved that in every dimension n there exists only a finite number of isomorphic crystallographic groups and also gave a description of these groups. From the perspective of differential geometry these results are of major importance, as crystallographic groups are precisely the fundamental groups of compact flat Riemannian orbifolds. The quotient is even a manifold if the fundamental group is required to be torsion-free, in which case it is called a Bieberbach group. Moreover, for a flat manifold the fundamental group completely determines the holonomy group. In this talk I will discuss the properties of crystallographic groups, study examples in dimension n=2 and n=3, and present the three Bieberbach theorems on the structure of crystallographic groups.
Coincidences
14:10 Mon 20 May, 2013 :: 7.15 Ingkarni Wardli :: A/Prof. Robb Muirhead :: School of Mathematical Sciences

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This is a lighthearted (some would say content-free) talk about coincidences, those surprising concurrences of events that are often perceived as meaningfully related, with no apparent causal connection. Time permitting, it will touch on topics like:
Patterns in data and the dangers of looking for patterns, unspecified ahead of time, and trying to "explain" them; e.g. post hoc subgroup analyses, cancer clusters, conspiracy theories ...
Matching problems; e.g. the birthday problem and extensions
People who win a lottery more than once -- how surprised should we really be? What's the question we should be asking?
When you become familiar with a new word, and see it again soon afterwards, how surprised should you be?
Caution: This is a shortened version of a talk that was originally prepared for a group of non-mathematicians and non-statisticians, so it's mostly non-technical. It probably does not contain anything you don't already know -- it will be an amazing coincidence if it does!
A strong Oka principle for proper immersions of finitely connected planar domains into CxC*
12:10 Fri 31 May, 2013 :: Ingkarni Wardli B19 :: Dr Tyson Ritter :: University of Adelaide

Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. In previous work we showed that, given a continuous map from X to the elliptic manifold CxC*, where X is a finitely connected planar domain without isolated boundary points, a stronger Oka property holds whereby the map is homotopic to a proper holomorphic embedding. If the planar domain is additionally permitted to have isolated boundary points the problem becomes more difficult, and it is not yet clear whether a strong Oka property for embeddings into CxC* continues to hold. We will discuss recent results showing that every continuous map from a finitely connected planar domain into CxC* is homotopic to a proper immersion that, in most cases, identifies at most finitely many pairs of distinct points. This is joint work with Finnur Larusson.
K-homology and the quantization commutes with reduction problem
12:10 Fri 5 Jul, 2013 :: 7.15 Ingkarni Wardli :: Prof Nigel Higson :: Pennsylvania State University

The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s using geometric techniques, and solved again shortly afterwards by Tian and Zhang using analytic methods. In this talk I shall outline some of the close links that exist between the problem, the two solutions, and the geometric and analytic versions of K-homology theory that are studied in noncommutative geometry. I shall try to make the case for K-homology as a useful conceptual framework for the solutions and (at least some of) their various generalizations.
The Hamiltonian Cycle Problem and Markov Decision Processes
15:10 Fri 2 Aug, 2013 :: B.18 Ingkarni Wardli :: Prof Jerzy Filar :: Flinders University

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We consider the famous Hamiltonian cycle problem (HCP) embedded in a Markov decision process (MDP). More specifically, we consider a moving object on a graph G where, at each vertex, a controller may select an arc emanating from that vertex according to a probabilistic decision rule. A stationary policy is simply a control where these decision rules are time invariant. Such a policy induces a Markov chain on the vertices of the graph. Therefore, HCP is equivalent to a search for a stationary policy that induces a 0-1 probability transition matrix whose non-zero entries trace out a Hamiltonian cycle in the graph. A consequence of this embedding is that we may consider the problem over a number of, alternative, convex - rather than discrete - domains. These include: (a) the space of stationary policies, (b) the more restricted but, very natural, space of doubly stochastic matrices induced by the graph, and (c) the associated spaces of so-called "occupational measures". This approach to the HCP has led to both theoretical and algorithmic approaches to the underlying HCP problem. In this presentation, we outline a selection of results generated by this line of research.
Privacy-Preserving Computation: Not just for secretive millionaires*
12:10 Mon 19 Aug, 2013 :: B.19 Ingkarni Wardli :: Wilko Henecka :: University of Adelaide

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PPC enables parties to share information while preserving their data privacy. I will introduce the concept, show a common ingredient and illustrate its use in an example. *See Yao's Millionaires Problem.
Random Wanderings on a Sphere...
11:10 Tue 17 Sep, 2013 :: Ingkarni Wardli Level 5 Room 5.57 :: A/Prof Robb Muirhead :: University of Adelaide

This will be a short talk (about 30 minutes) about the following problem. (Even if I tell you all I know about it, it won't take very long!) Imagine the earth is a unit sphere in 3-dimensions. You're standing at a fixed point, which we may as well take to be the North Pole. Suddenly you get moved to another point on the sphere by a random (uniform) orthogonal transormation. Where are you now? You're not at a point which is uniformly distributed on the surface of the sphere (so, since most of the earth's surface is water, you're probably drowning). But then you get moved again by the same orthogonal transformation. Where are you now? And what happens to your location it this happens repeatedly? I have only a partial answwer to this question, for 2 and 3 transformations. (There's nothing special about 3 dimensions here--results hold for all dimensions which are at least 3.) I don't know of any statistical application for this! This work was motivated by a talk I heard, given by Tom Marzetta (Bell Labs) at a conference at MIT. Although I know virtually nothing about signal processing, I gather Marzetta was trying to encode signals using powers of ranfom orthogonal matrices. After carrying out simulations, I think he decided it wasn't a good idea.
Symmetry gaps for geometric structures
15:10 Fri 20 Sep, 2013 :: B.18 Ingkarni Wardli :: Dr Dennis The :: Australian National University

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Klein's Erlangen program classified geometries based on their (transitive) groups of symmetries, e.g. Euclidean geometry is the quotient of the rigid motion group by the subgroup of rotations. While this perspective is homogeneous, Riemann's generalization of Euclidean geometry is in general very "lumpy" - i.e. there exist Riemannian manifolds that have no symmetries at all. A common generalization where a group still plays a dominant role is Cartan geometry, which first arose in Cartan's solution to the equivalence problem for geometric structures, and which articulates what a "curved version" of a flat (homogeneous) model means. Parabolic geometries are Cartan geometries modelled on (generalized) flag varieties (e.g. projective space, isotropic Grassmannians) which are well-known objects from the representation theory of semisimple Lie groups. These curved versions encompass a zoo of interesting geometries, including conformal, projective, CR, systems of 2nd order ODE, etc. This interaction between differential geometry and representation theory has proved extremely fruitful in recent years. My talk will be an example-based tour of various types of parabolic geometries, which I'll use to outline some of the main aspects of the theory (suppressing technical details). The main thread throughout the talk will be the symmetry gap problem: For a given type of Cartan geometry, the maximal symmetry dimension is realized by the flat model, but what is the next possible ("submaximal") symmetry dimension? I'll sketch a recent solution (in joint work with Boris Kruglikov) for a wide class of parabolic geometries which gives a combinatorial recipe for reading the submaximal symmetry dimension from a Dynkin diagram.
How to stack oranges in three dimensions, 24 dimensions and beyond
18:00 Thu 26 Sep, 2013 :: Horace Lamb Lecture Theatre :: Prof Akshay Venkatesh :: Stanford University

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How can we pack balls as tightly as possible? In other words: to squeeze as many balls as possible into a limited space, what's the best way of arranging the balls? It's not hard to guess what the answer should be - but it's very hard to be sure that it really is the answer! I'll tell the interesting story of this problem, going back to the astronomer Kepler, and ending almost four hundred years later with Thomas Hales. I will then talk about stacking 24-dimensional oranges: what this means, how it relates to the Voyager spacecraft, and the many things we don't know beyond this.
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.
Group meeting
15:10 Fri 25 Oct, 2013 :: 5.58 (Ingkarni Wardli) :: Dr Ben Binder and Mr David Wilke :: University of Adelaide

Dr Ben Binder :: 'An inverse approach for solutions to free-surface flow problems' :: Abstract: Surface water waves are familiar to most people, for example, the wave pattern generated at the stern of a ship. The boundary or interface between the air and water is called the free-surface. When determining a solution to a free-surface flow problem it is commonplace for the forcing (eg. shape of ship or waterbed topography) that creates the surface waves to be prescribed, with the free-surface coming as part of the solution. Alternatively, one can choose to prescribe the shape of the free-surface and find the forcing inversely. In this talk I will discuss my ongoing work using an inverse approach to discover new types of solutions to free-surface flow problems in two and three dimensions, and how the predictions of the method might be verified with experiments. :: Mr David Wilke:: 'A Computational Fluid Dynamic Study of Blood Flow Within the Coiled Umbilical Arteries':: Abstract: The umbilical cord is the lifeline of the fetus throughout gestation. In a normal pregnancy it facilitates the supply of oxygen and nutrients from the placenta via a single vein, in addition to the return of deoxygenated blood from the developing embryo or fetus via two umbilical arteries. Despite the major role it plays in the growth of the fetus, pathologies of the umbilical cord are poorly understood. In particular, variations in the cord geometry, which typically forms a helical arrangement, have been correlated with adverse outcomes in pregnancy. Cords exhibiting either abnormally low or high levels of coiling have been associated with pathological results including growth-restriction and fetal demise. Despite this, the methodology currently employed by clinicians to characterise umbilical pathologies can misdiagnose cords and is prone to error. In this talk a computational model of blood flow within rigid three-dimensional structures representative of the umbilical arteries will be presented. This study determined that the current characterization was unable to differentiate between cords which exhibited clinically distinguishable flow properties, including the cord pressure drop, which provides a measure of the loading on the fetal heart.
The geometry of rolling surfaces and non-holonomic mechanics
15:10 Fri 1 Nov, 2013 :: B.18 Ingkarni Wardli :: Prof Robert Bryant :: Duke University

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In mechanics, the system of a sphere rolling over a plane without slipping or twisting is a fundamental example of what is called a non-holonomic mechanical system, the study of which belongs to the subject of control theory. The more general case of one surface rolling over another without slipping or twisting is, similarly, of great interest for both practical and theoretical reasons. In this talk, which is intended for a general mathematical audience (i.e., no familiarity with control theory or differential geometry will be assumed), I will describe some of the basic features of this problem, a bit of its history, and some of the surprising developments that its study reveals, such as the unexpected appearance of the exceptional group G_2.
Braids, conformal module and entropy
12:10 Fri 8 Nov, 2013 :: Ingkarni Wardli B19 :: Prof Burglind Joricke :: Australian National University

I will discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which implicitly occurred already in a paper of Gorin and Lin in connection with their interest in Hilbert's 13th problem. The second is a popular dynamical invariant, the entropy. It appeared in connection with Thurston's theory of surface homeomorphisms. It turns out that these invariants are related: They are inversely proportional. In a preparatory talk (at 10:10 am) I will give a brief introduction to some aspects of braid theory and to entropy.
Euler and Lagrange solutions of the three-body problem and beyond
12:10 Fri 15 Nov, 2013 :: Ingkarni Wardli B19 :: Prof Pawel Nurowski :: Centre for Theoretical Physics, Polish Academy of Sciences

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.)
Viscoelastic fluids: mathematical challenges in determining their relaxation spectra
15:10 Mon 17 Mar, 2014 :: 5.58 Ingkarni Wardli :: Professor Russell Davies :: Cardiff University

Determining the relaxation spectrum of a viscoelastic fluid is a crucial step before a linear or nonlinear constitutive model can be applied. Information about the relaxation spectrum is obtained from simple flow experiments such as creep or oscillatory shear. However, the determination process involves the solution of one or more highly ill-posed inverse problems. The availability of only discrete data, the presence of noise in the data, as well as incomplete data, collectively make the problem very hard to solve. In this talk I will illustrate the mathematical challenges inherent in determining relaxation spectra, and also introduce the method of wavelet regularization which enables the representation of a continuous relaxation spectrum by a set of hyperbolic scaling functions.
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.
The p-Minkowski problem
12:10 Fri 13 Jun, 2014 :: Ingkarni Wardli B20 :: Xu-Jia Wang :: Australian National University

The p-Minkowski problem is an extension of the classical Minkowski problem. It concerns the existence, uniqueness, and regularity of closed convex hypersurfaces with prescribed Gauss curvature. The Minkowski problem has been studied by many people in the last century and has been completely resolved. The p-Minkowski problem involves more applications. In this talk we will review the development of the study of the p-Minkowski problem and discuss some recent works on the problem.​
Optimal transportation and Monge-Ampere type equation
15:10 Fri 13 Jun, 2014 :: B.21 Ingkarni Wardli :: Professor Xu-Jia Wang :: Centre for Mathematics and its Applications, Australian National University

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The optimal transportation is to find an optimal mapping of transferring one mass density to another one such that the total cost is minimised. This problem was first introduced by Monge in 1781. Monge's cost function is propositional to the distance the mass is transferred, namely c(x,y)=|x-y|, but more general costs are allowed. The optimal transportation has found a variety of applications and has been extensively studied since then. In 1940s Kantorovich introduced a dual functional, by which one can determine the optimal mapping through the associated potential function, for a large class of cost functions. The potential function satisfies a Monge-Ampere type equation, which is a fully nonlinear partial differential equation arising also in geometric problems related to the Gauss curvature, and has been studied by Aleksandrov, Calabi, Nirenberg, Pogorelov, Cheng-Yau, and Caffarelli, among many others. In this talk we will first introduce the optimal transportation and review the existence of optimal mappings. We then focus on the regularity of the optimal mappings. By studying the associated Monge-Ampere equation, sharp conditions on the cost function have been found by the speaker and his collaborators. For Monge's cost function |x-y|, which does not satisfy the sharp conditions, we have also obtained the existence of optimal mappings, and established interesting regularity and singularity results for the mapping.
The Dirichlet problem for the prescribed Ricci curvature equation
12:10 Fri 15 Aug, 2014 :: Ingkarni Wardli B20 :: Artem Pulemotov :: University of Queensland

We will discuss the following question: is it possible to find a Riemannian metric whose Ricci curvature is equal to a given tensor on a manifold M? To answer this question, one must analyze a weakly elliptic second-order geometric PDE. In the first part of the talk, we will review the history of the subject and state several classical theorems. After that, our focus will be on new results concerning the case where M has nonempty boundary.
Boundary-value problems for the Ricci flow
15:10 Fri 15 Aug, 2014 :: B.18 Ingkarni Wardli :: Dr Artem Pulemotov :: The University of Queensland

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The Ricci flow is a differential equation describing the evolution of a Riemannian manifold (i.e., a "curved" geometric object) into an Einstein manifold (i.e., an object with a "constant" curvature). This equation is particularly famous for its key role in the proof of the Poincare Conjecture. Understanding the Ricci flow on manifolds with boundary is a difficult problem with applications to a variety of fields, such as topology and mathematical physics. The talk will survey the current progress towards the resolution of this problem. In particular, we will discuss new results concerning spaces with symmetries.
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.
Exploration vs. Exploitation with Partially Observable Gaussian Autoregressive Arms
15:00 Mon 29 Sep, 2014 :: Engineering North N132 :: Julia Kuhn :: The University of Queensland & The University of Amsterdam

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We consider a restless bandit problem with Gaussian autoregressive arms, where the state of an arm is only observed when it is played and the state-dependent reward is collected. Since arms are only partially observable, a good decision policy needs to account for the fact that information about the state of an arm becomes more and more obsolete while the arm is not being played. Thus, the decision maker faces a tradeoff between exploiting those arms that are believed to be currently the most rewarding (i.e. those with the largest conditional mean), and exploring arms with a high conditional variance. Moreover, one would like the decision policy to remain tractable despite the infinite state space and also in systems with many arms. A policy that gives some priority to exploration is the Whittle index policy, for which we establish structural properties. These motivate a parametric index policy that is computationally much simpler than the Whittle index but can still outperform the myopic policy. Furthermore, we examine the many-arm behavior of the system under the parametric policy, identifying equations describing its asymptotic dynamics. Based on these insights we provide a simple heuristic algorithm to evaluate the performance of index policies; the latter is used to optimize the parametric index.
Optimally Chosen Quadratic Forms for Partitioning Multivariate Data
13:10 Tue 14 Oct, 2014 :: Ingkarni Wardli 715 Conference Room :: Assoc. Prof. Inge Koch :: School of Mathematical Sciences

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Quadratic forms are commonly used in linear algebra. For d-dimensional vectors they have a matrix representation, Q(x) = x'Ax, for some symmetric matrix A. In statistics quadratic forms are defined for d-dimensional random vectors, and one of the best-known quadratic forms is the Mahalanobis distance of two random vectors. In this talk we want to partition a quadratic form Q(X) = X'MX, where X is a random vector, and M a symmetric matrix, that is, we want to find a d-dimensional random vector W such that Q(X) = W'W. This problem has many solutions. We are interested in a solution or partition W of X such that pairs of corresponding variables (X_j, W_j) are highly correlated and such that W is simpler than the given X. We will consider some natural candidates for W which turn out to be suboptimal in the sense of the above constraints, and we will then exhibit the optimal solution. Solutions of this type are useful in the well-known T-square statistic. We will see in examples what these solutions look like.
Happiness and social information flow: Computational social science through data.
15:10 Fri 7 Nov, 2014 :: EM G06 (Engineering & Maths Bldg) :: Dr Lewis Mitchell :: University of Adelaide

The recent explosion in big data coming from online social networks has led to an increasing interest in bringing quantitative methods to bear on questions in social science. A recent high-profile example is the study of emotional contagion, which has led to significant challenges and controversy. This talk will focus on two issues related to emotional contagion, namely remote-sensing of population-level wellbeing and the problem of information flow across a social network. We discuss some of the challenges in working with massive online data sets, and present a simple tool for measuring large-scale happiness from such data. By combining over 10 million geolocated messages collected from Twitter with traditional census data we uncover geographies of happiness at the scale of states and cities, and discuss how these patterns may be related to traditional wellbeing measures and public health outcomes. Using tools from information theory we also study information flow between individuals and how this may relate to the concept of predictability for human behaviour.
Happiness and social information flow: Computational social science through data.
15:10 Fri 7 Nov, 2014 :: EM G06 (Engineering & Maths Bldg) :: Dr Lewis Mitchell :: University of Adelaide

The recent explosion in big data coming from online social networks has led to an increasing interest in bringing quantitative methods to bear on questions in social science. A recent high-profile example is the study of emotional contagion, which has led to significant challenges and controversy. This talk will focus on two issues related to emotional contagion, namely remote-sensing of population-level wellbeing and the problem of information flow across a social network. We discuss some of the challenges in working with massive online data sets, and present a simple tool for measuring large-scale happiness from such data. By combining over 10 million geolocated messages collected from Twitter with traditional census data we uncover geographies of happiness at the scale of states and cities, and discuss how these patterns may be related to traditional wellbeing measures and public health outcomes. Using tools from information theory we also study information flow between individuals and how this may relate to the concept of predictability for human behaviour.
Multivariate regression in quantitative finance: sparsity, structure, and robustness
15:10 Fri 1 May, 2015 :: Engineering North N132 :: A/Prof Mark Coates :: McGill University

Many quantitative hedge funds around the world strive to predict future equity and futures returns based on many sources of information, including historical returns and economic data. This leads to a multivariate regression problem. Compared to many regression problems, the signal-to-noise ratio is extremely low, and profits can be realized if even a small fraction of the future returns can be accurately predicted. The returns generally have heavy-tailed distributions, further complicating the regression procedure.

In this talk, I will describe how we can impose structure into the regression problem in order to make detection and estimation of the very weak signals feasible. Some of this structure consists of an assumption of sparsity; some of it involves identification of common factors to reduce the dimension of the problem. I will also describe how we can formulate alternative regression problems that lead to more robust solutions that better match the performance metrics of interest in the finance setting.

Can mathematics help save energy in computing?
15:10 Fri 22 May, 2015 :: Engineering North N132 :: Prof Markus Hegland :: ANU

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Recent development of computational hardware is characterised by two trends: 1. High levels of duplication of computational capabilities in multicore, parallel and GPU processing, and, 2. Substantially faster development of the speed of computational technology compared to communication technology

A consequence of these two trends is that energy costs of modern computing devices from mobile phones to supercomputers are increasingly dominated by communication costs. In order to save energy one would thus need to reduce the amount of data movement within the computer. This can be achieved by recomputing results instead of communicating them. The resulting increase in computational redundancy may also be used to make the computations more robust against hardware faults. Paradoxically, by doing more (computations) we do use less (energy).

This talk will first discuss for a simple example how a mathematical understanding can be applied to improve computational results using extrapolation. Then the problem of energy consumption in computational hardware will be considered. Finally some recent work will be discussed which shows how redundant computing is used to mitigate computational faults and thus to save energy.

Monodromy of the Hitchin system and components of representation varieties
12:10 Fri 29 May, 2015 :: Napier 144 :: David Baraglia :: University of Adelaide

Representations of the fundamental group of a compact Riemann surface into a reductive Lie group form a moduli space, called a representation variety. An outstanding problem in topology is to determine the number of components of these varieties. Through a deep result known as non-abelian Hodge theory, representation varieties are homeomorphic to moduli spaces of certain holomorphic objects called Higgs bundles. In this talk I will describe recent joint work with L. Schaposnik computing the monodromy of the Hitchin fibration for Higgs bundle moduli spaces. Our results give a new unified proof of the number of components of several representation varieties.
Group Meeting
15:10 Fri 29 May, 2015 :: EM 213 :: Dr Judy Bunder :: University of Adelaide

Talk : Patch dynamics for efficient exascale simulations Abstract Massive parallelisation has lead to a dramatic increase in available computational power. However, data transfer speeds have failed to keep pace and are the major limiting factor in the development of exascale computing. New algorithms must be developed which minimise the transfer of data. Patch dynamics is a computational macroscale modelling scheme which provides a coarse macroscale solution of a problem defined on a fine microscale by dividing the domain into many nonoverlapping, coupled patches. Patch dynamics is readily adaptable to massive parallelisation as each processor core can evaluate the dynamics on one, or a few, patches. However, patch coupling conditions interpolate across the unevaluated parts of the domain between patches and require almost continuous data transfer. We propose a modified patch dynamics scheme which minimises data transfer by only reevaluating the patch coupling conditions at `mesoscale' time scales which are significantly larger than the microscale time of the microscale problem. We analyse and quantify the error arising from patch dynamics with mesoscale temporal coupling.
Gromov's method of convex integration and applications to minimal surfaces
12:10 Fri 7 Aug, 2015 :: Ingkarni Wardli B17 :: Finnur Larusson :: The University of Adelaide

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We start by considering an applied problem. You are interested in buying a used car. The price is tempting, but the car has a curious defect, so it is not clear whether you can even take it for a test drive. This problem illustrates the key idea of Gromov's method of convex integration. We introduce the method and some of its many applications, including new applications in the theory of minimal surfaces, and end with a sketch of ongoing joint work with Franc Forstneric.
Bilinear L^p estimates for quasimodes
12:10 Fri 14 Aug, 2015 :: Ingkarni Wardli B17 :: Melissa Tacy :: The University of Adelaide

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Understanding the growth of the product of eigenfunctions $$u\cdot{}v$$ $$\Delta{}u=-\lambda^{2}u\quad{}\Delta{}v=-\mu^{2}v$$ is vital to understanding the regularity properties of non-linear PDE such as the non-linear Schr\"{o}dinger equation. In this talk I will discuss some recent results that I have obtain in collaboration with Zihua Guo and Xiaolong Han which provide a full range of estimates of the form $$||uv||_{L^{p}}\leq{}G(\lambda,\mu)||u||_{L^{2}}||v||_{L^{2}}$$ where $u$ and $v$ are approximate eigenfunctions of the Laplacian. We obtain these results by re-casting the problem to a more general related semiclassical problem.
The Calderon Problem: From the Past to the Present
15:10 Fri 11 Sep, 2015 :: Ingkarni Wardli B21 :: Dr Leo Tzou :: University of Sydney

The problem of determining the electrical conductivity of a body by making voltage and current measurements on the object's surface has various applications in fields such as oil exploration and early detection of malignant breast tumour. This classical problem posed by Calderon remained open until the late '80s when it was finally solved in a breakthrough paper by Sylvester-Uhlmann.

In the recent years, geometry has played an important role in this problem. The unexpected connection of this subject to fields such as dynamical systems, symplectic geometry, and Riemannian geometry has led to some interesting progress. This talk will be an overview of some of the recent results and an outline of the techniques used to treat this problem.

Chern-Simons classes on loop spaces and diffeomorphism groups
12:10 Fri 16 Oct, 2015 :: Ingkarni Wardli B17 :: Steve Rosenberg :: Boston University

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Not much is known about the topology of the diffeomorphism group Diff(M) of manifolds M of dimension four and higher. We'll show that for a class of manifolds of dimension 4k+1, Diff(M) has infinite fundamental group. This is proved by translating the problem into a question about Chern-Simons classes on the tangent bundle to the loop space LM. To build the CS classes, we use a family of metrics on LM associated to a Riemannian metric on M. The curvature of these metrics takes values in an algebra of pseudodifferential operators. The main technical step in the CS construction is to replace the ordinary matrix trace in finite dimensions with the Wodzicki residue, the unique trace on this algebra. The moral is that some techniques in finite dimensional Riemannian geometry can be extended to some examples in infinite dimensional geometry.
Group meeting
15:10 Fri 20 Nov, 2015 :: Ingkarni Wardli B17 :: Mr Jack Keeler :: University of East Anglia / University of Adelaide

Title: Stability of free-surface flow over topography Abstract: The forced KdV equation is used as a model to analyse the wave behaviour on the free surface in response to prescribed topographic forcing. The research involves computing steady solutions using numeric and asymptotic techniques and then analysing the stability of these steady solutions in time-dependent calculations. Stability is analysed by computing the eigenvalue spectra of the linearised fKdV operator and by exploiting the Hamiltonian structure of the fKdV. Future work includes analysing the solution space for a corrugated topography and investigating the 3 dimensional problem using the KP equation. + Any items for group discussion
Group meeting
15:10 Fri 20 Nov, 2015 :: Ingkarni Wardli B17 :: Mr Jack Keeler :: University of East Anglia / University of Adelaide

Title: Stability of free-surface flow over topography Abstract: The forced KdV equation is used as a model to analyse the wave behaviour on the free surface in response to prescribed topographic forcing. The research involves computing steady solutions using numeric and asymptotic techniques and then analysing the stability of these steady solutions in time-dependent calculations. Stability is analysed by computing the eigenvalue spectra of the linearised fKdV operator and by exploiting the Hamiltonian structure of the fKdV. Future work includes analysing the solution space for a corrugated topography and investigating the 3 dimensional problem using the KP equation. + Any items for group discussion
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.
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.
Extreme eigenvalues
15:10 Fri 29 Apr, 2016 :: Engineering South S112 :: Dr Julie Clutterbuck :: Monash University

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Each bounded domain has a sequence of eigenvalues attached to it. These are determined by the geometry of the domain, but do not completely encode the geometry. A natural question is to ask: which domains optimise the eigenvalues? For example, which domains have the smallest or largest first eigenvalue, or have the largest gap between eigenvalues? This is a rather old problem, with connections to the isoperimetric problem. I will describe some old and new results.
Behavioural Microsimulation Approach to Social Policy and Behavioural Economics
15:10 Fri 20 May, 2016 :: S112 Engineering South :: Dr Drew Mellor :: Ernst & Young

SIMULAIT is a general purpose, behavioural micro-simulation system designed to predict behavioural trends in human populations. This type of predictive capability grew out of original research initially conducted in conjunction with the Defence Science and Technology Group (DSTO) in South Australia, and has been fully commercialised and is in current use by a global customer base. To our customers, the principal value of the system lies in its ability to predict likely outcomes to scenarios that challenge conventional approaches based on extrapolation or generalisation. These types of scenarios include: the impact of disruptive technologies, such as the impact of wide-spread adoption of autonomous vehicles for transportation or batteries for household energy storage; and the impact of effecting policy elements or interventions, such as the impact of imposing water usage restrictions. SIMULAIT employs a multi-disciplinary methodology, drawing from agent-based modelling, behavioural science and psychology, microeconomics, artificial intelligence, simulation, game theory, engineering, mathematics and statistics. In this seminar, we start with a high-level view of the system followed by a look under the hood to see how the various elements come together to answer questions about behavioural trends. The talk will conclude with a case study of a recent application of SIMULAIT to a significant policy problem - how to address the deficiency of STEM skilled teachers in the Victorian teaching workforce.
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.
Predicting turbulence
14:10 Tue 30 Aug, 2016 :: Napier 209 :: Dr Trent Mattner :: School of Mathematical Sciences

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Turbulence is characterised by three-dimensional unsteady fluid motion over a wide range of spatial and temporal scales. It is important in many problems of technological and scientific interest, such as drag reduction, energy production and climate prediction. Turbulent flows are governed by the Navier--Stokes equations, which are a nonlinear system of partial differential equations. Typically, numerical methods are needed to find solutions to these equations. In turbulent flows, however, the resulting computational problem is usually intractable. Filtering or averaging the Navier--Stokes equations mitigates the computational problem, but introduces new quantities into the equations. Mathematical models of turbulence are needed to estimate these quantities. One promising turbulence model consists of a random collection of fluid vortices, which are themselves approximate solutions of the Navier--Stokes equations.
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.
Fault tolerant computation of hyperbolic PDEs with the sparse grid combination technique
15:10 Fri 28 Oct, 2016 :: Ingkarni Wardli 5.57 :: Dr Brendan Harding :: University of Adelaide

Computing solutions to high dimensional problems is challenging because of the curse of dimensionality. The sparse grid combination technique allows one to significantly reduce the cost of computing solutions such that they become manageable on current supercomputers. However, as these supercomputers increase in size the rate of failure also increases. This poses a challenge for our computations. In this talk we look at the problem of computing solutions to hyperbolic partial differential equations with the combination technique in an environment where faults occur. A fault tolerant generalisation of the combination technique will be presented along with results that demonstrate its effectiveness.
Plumbing regular closed polygonal curves
12:10 Mon 22 May, 2017 :: Inkgarni Wardli Conference Room 715 :: Dr Barry Cox :: School of Mathematical Sciences

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In 1980 the following puzzle appeared in Mathematics Magazine: A certain mathematician, in order to make ends meet, moonlights as an apprentice plumber. One night, as the mathematician contemplated a pile of straight pipes of equal lengths and right-angled elbows, the following question occurred to this mathematician: ``For which positive integers n could I form a closed polygonal curve using n such straight pipes and n elbows?'' It turns out that it is possible for any even number n greater than or equal to 4 and any odd number n greater than or equal to 7. However the case n=7 is particularly interesting because it can be done one of two ways and the problem is related to that of determining all the possible conformations of the molecule cyclo-heptane, although the angles in cyclo-heptane are not right angles. This raises the questions: ``Do the two solutions to the maths puzzle with right-angles correspond to the two principal conformations of cyclo-heptane?'', and ``How many solutions/conformations exist for other elbow angles?'' These and other issues will be discussed.
Stokes' Phenomenon in Translating Bubbles
15:10 Fri 2 Jun, 2017 :: Ingkarni Wardli 5.57 :: Dr Chris Lustri :: Macquarie University

This study of translating air bubbles in a Hele-Shaw cell containing viscous fluid reveals the critical role played by surface tension in these systems. The standard zero-surface-tension model of Hele-Shaw flow predicts that a continuum of bubble solutions exists for arbitrary flow translation velocity. The inclusion of small surface tension, however, eliminates this continuum of solutions, instead producing a discrete, countably infinite family of solutions, each with distinct translation speeds. We are interested in determining this discrete family of solutions, and understanding why only these solutions are permitted. Studying this problem in the asymptotic limit of small surface tension does not seem to give any particular reason why only these solutions should be selected. It is only by using exponential asymptotic methods to study the Stokes’ structure hidden in the problem that we are able to obtain a complete picture of the bubble behaviour, and hence understand the selection mechanism that only permits certain solutions to exist. In the first half of my talk, I will explain the powerful ideas that underpin exponential asymptotic techniques, such as analytic continuation and optimal truncation. I will show how they are able to capture behaviour known as Stokes' Phenomenon, which is typically invisible to classical asymptotic series methods. In the second half of the talk, I will introduce the problem of a translating air bubble in a Hele-Shaw cell, and show that the behaviour can be fully understood by examining the Stokes' structure concealed within the problem. Finally, I will briefly showcase other important physical applications of exponential asymptotic methods, including submarine waves and particle chains.
Topology as a tool in algebra
15:10 Fri 8 Sep, 2017 :: Ingkarni Wardli B17 :: Dr Zsuzsanna Dancso :: University of Sydney

Topologists often use algebra in order to understand the shape of a space: invariants such as homology and cohomology are basic, and very successful, examples of this principle. Although topology is used as a tool in algebra less often, I will describe a recurring pattern on the border of knot theory and quantum algebra where this is possible. We will explore how the tangled topology of "flying circles in R^3" is deeply related to a famous problem in Lie theory: the Kashiwara-Vergne (KV) problem (first solved in 2006 by Alekseev-Meinrenken). I will explain how this relationship illuminates the intricate algebra of the KV problem.
Equivariant formality of homogeneous spaces
12:10 Fri 29 Sep, 2017 :: Engineering Sth S111 :: Alex Chi-Kwong Fok :: University of Adelaide

Equivariant formality, a notion in equivariant topology introduced by Goresky-Kottwitz-Macpherson, is a desirable property of spaces with group actions, which allows the application of localisation formula to evaluate integrals of any top closed forms and enables one to compute easily the equivariant cohomology. Broad classes of spaces of especial interest are well-known to be equivariantly formal, e.g., compact symplectic manifolds equipped with Hamiltonian compact Lie group actions and projective varieties equipped with linear algebraic torus actions, of which flag varieties are examples. Less is known about compact homogeneous spaces G/K equipped with the isotropy action of K, which is not necessarily of maximal rank. In this talk we will review previous attempts of characterizing equivariant formality of G/K, and present our recent results on this problem using an analogue of equivariant formality in K-theory. Part of the work presented in this talk is joint with Jeffrey Carlson.
A multiscale approximation of a Cahn-Larche system with phase separation on the microscale
15:10 Thu 22 Feb, 2018 :: Ingkarni Wardli 5.57 :: Ms Lisa Reischmann :: University of Augsberg

We consider the process of phase separation of a binary system under the influence of mechanical deformation and we derive a mathematical multiscale model, which describes the evolving microstructure taking into account the elastic properties of the involved materials. Motivated by phase-separation processes observed in lipid monolayers in film-balance experiments, the starting point of the model is the Cahn-Hilliard equation coupled with the equations of linear elasticity, the so-called Cahn-Larche system. Owing to the fact that the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level, a multiscale approach is imperative. We assume the pattern of the evolving microstructure to have an intrinsic length scale associated with it, which, after nondimensionalisation, leads to a scaled model involving a small parameter epsilon>0, which is suitable for periodic-homogenisation techniques. For the full nonlinear problem the so-called homogenised problem is then obtained by letting epsilon tend to zero using the method of asymptotic expansion. Furthermore, we present a linearised Cahn-Larche system and use the method of two-scale convergence to obtain the associated limit problem, which turns out to have the same structure as in the nonlinear case, in a mathematically rigorous way. Properties of the limit model will be discussed.
Stability Through a Geometric Lens
15:10 Fri 18 May, 2018 :: Horace Lamb 1022 :: Dr Robby Marangell :: University of Sydney

Focussing on the example of the Fisher/KPP equation, I will show how geometric information can be used to establish (in)stability results in some partial differential equations (PDEs). Viewing standing and travelling waves as fixed points of a flow in an infinite dimensional system, leads to a reduction of the linearised stability problem to a boundary value problem in a linear non-autonomous ordinary differential equation (ODE). Next, by exploiting the linearity of the system, one can use geometric ideas to reveal additional structure underlying the determination of stability. I will show how the Riccati equation can be used to produce a reasonably computable detector of eigenvalues and how such a detector is related to another, well-known eigenvalue detector, the Evans function. If there is time, I will try to expand on how to generalise these ideas to systems of PDEs.
Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory
13:10 Fri 25 May, 2018 :: Barr Smith South Polygon Lecture theatre :: David Baraglia :: University of Adelaide

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Let X be a smooth, compact, oriented 4-manifold and consider the following problem. Let G be a group which acts on the second cohomology of X preserving the intersection form. Can this action of G on H^2(X) be lifted to an action of G on X by diffeomorphisms? We study a parametrised version of Seiberg-Witten theory for smooth families of 4-manifolds and obtain obstructions to the existence of such lifts. For example, we construct compact simply-connected 4-manifolds X and involutions on H^2(X) that can be realised by a continuous involution on X, or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on X.
Hitchin's Projectively Flat Connection for the Moduli Space of Higgs Bundles
13:10 Fri 15 Jun, 2018 :: Barr Smith South Polygon Lecture theatre :: John McCarthy :: University of Adelaide

In this talk I will discuss the problem of geometrically quantizing the moduli space of Higgs bundles on a compact Riemann surface using Kahler polarisations. I will begin by introducing geometric quantization via Kahler polarisations for compact manifolds, leading up to the definition of a Hitchin connection as stated by Andersen. I will then describe the moduli spaces of stable bundles and Higgs bundles over a compact Riemann surface, and discuss their properties. The problem of geometrically quantizing the moduli space of stables bundles, a compact space, was solved independently by Hitchin and Axelrod, Del PIetra, and Witten. The Higgs moduli space is non-compact and therefore the techniques used do not apply, but carries an action of C*. I will finish the talk by discussing the problem of finding a Hitchin connection that preserves this C* action. Such a connection exists in the case of Higgs line bundles, and I will comment on the difficulties in higher rank.

News matching "The two envelope problem"

Outstanding results in the COMAP Mathematical Contest in Modeling
Congratulations to Parsa Kavkani, Alex Tam, Leon Chea, Helen Geng and Susan Pang, who participated in this year's Mathematical Contest in Modeling, run by the Consortium for Mathematics and Its Applications (COMAP). The team with Parsa Kavkani and Alex Tam was designated an "Outstanding Winner" for Problem A (on the spreading of Ebola) and was awarded an INFORMS award for their work. Only 5 outstanding winners were selected from over 5000 entries for this problem, which is an amazing achievement. The team with Leon Chea, Helen Geng and Susan Pang was designated a "Meritorious Winner" for Problem A. There were about 640 meritorious winners out of the 5000, which is also an excellent achievement. Posted Tue 28 Apr 15.

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Publications matching "The two envelope problem"

Publications
A discrete version of the Riemann Hilbert problem
Larusson, Finnur; Sadykov, T, Russian Mathematical Surveys 63 (973–975) 2008
The solution of a free boundary problem related to environmental management systems
Elliott, Robert; Filinkov, Alexei, Stochastic Analysis and Applications 25 (1189–1202) 2007
On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials
Clements, David; Ang, W, International Journal of Engineering Science 44 (273–284) 2006
Large-Reynolds-number asymptotics of the Berman problem
Cox, Stephen; King, J, Studies in Applied Mathematics 113 (217–243) 2004
The envelope of a one-dimensional pattern in the presence of a conserved quantity
Cox, Stephen, Physics Letters A 333 (91–101) 2004
On a convexity problem arising in queueing theory and electromagnetism
Peake, M; Pearce, Charles, Sixth International Conference on Nonlinear Functional Analysis and Applications, Gyeongsang National University 01/09/00
On an extremal problem arising in queueing theory and telecommunications
Peake, M; Pearce, Charles, chapter in Optimization and Related Topics (Kluwer Academic Publishers) 119–134, 2001

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