Search the School of Mathematical Sciences

It is well known that this construction is impossible, but an interesting question is whether this can be achieved to arbitrary accuracy. We show how this can be done. This construction generalizes to dividing angles into five equal parts, and so on.

Most students of high school mathematics will have encountered the technique of fitting a line to data by least squares. Those who have taken a university statistics course will also have heard this method referred to as regression. However, it is not obvious from common dictionary definitions why this should be the case. For example, "reversion to an earlier or less advanced state or form". In this talk, the mathematical phenomenon that gave regression its name will be explained and will be shown to have implications in some unexpected contexts.

Stirring and mixing of fluids occurs everywhere, from adding milk to a cup of coffee, right through to industrial-scale chemical blending. So why stir in the first place? Is it possible to do it badly? And how can you make sure you do it effectively? I will attempt to answer these questions using a few thought experiments, some dynamical systems theory and a little topology.

How often have you asked yourself: "I wonder how many fish are in that lake?" Probably never, but if you ever did, then this is the lecture for you. The solution is easy (Seuss, 1960), but raises the question of how good the answer is. I will answer this by looking at confidence intervals. In the lecture, I will discuss what a confidence interval is and how to calculate it using techniques for calculating probabilities in poker. I will also look at how these ideas have been used in epidemiology, the study of disease, to estimate the number of people with diabetes. [1] Seuss, Dr. (1960). "One Fish Two Fish Red Fish Blue Fish". Random House Books.

'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.

The job of the mathematician is to discover new truths about mathematical objects and their relationships. Such truths are established by proving them. This raises a fundamental question. Can every mathematical truth be proved (by a sufficiently clever being) or are there truths that will forever lie beyond the reach of proof? Mathematics can be turned on itself to investigate this question. In this talk, we will see that under certain assumptions about proofs, there are truths that cannot be proved. You must decide for yourself whether you think these assumptions are valid!

So far this year in the undergraduate seminars, we have seen how mathematics is useful for solving important problems, and how mathematics can be used to uncover profound truths. In this seminar I will show you something about prime numbers that is neither useful nor profound. I just think it is extremely cool.

The humble logarithm is a well known and dependable beast. In this talk we will provide a "greatest hits-tory" of the logarithm, highlighting some memorable moments from its first 400 years of life, and pondering some of the reasons why logarithms arise in so many diverse and unexpected situations. Finally, we will juggle some simple numerical coincidences to calculate a few choice logarithms from scratch.

Most people will have heard of IVF (in vitro fertilisation), a technology for helping infertile couples have a baby. Although there are many IVF babies, many will also know that the success rate is still low for the cost and inconvenience involved. The fact that some women cannot make use of IVF because of life-threatening consequences is less well known but motivates research into other technologies, including IVM (in vitro maturation). What has all this to do with maths? Come along and find out how mathematical modelling is contributing to understanding and improvement in this important and interesting field.

A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated to this equation is obtained, also by reduction.

We will report on our attempts to understand noncommutative geometry in the lights of the example of quantum spheres. We will see how to produce an equivariant fundamental class and also indicate some of the limitations of isospectral deformations.

This talk is concerned with the evolution of an unbounded inviscid fluid-fluid interface subject to an axisymmetric impulse in pressure and how inertial, interfacial and gravitational forces affect that evolution. The construct was motivated by the occurrence of lung hemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface allowing them to fill with blood. The evolution of the free surface is described by a boundary integral formulation which is integrated forward in time numerically. As the interface evolves, it is seen, depending upon the levels of gravity and surface tension, to form either axisymmetric surface jets, waves or droplets. Moreover the droplets may be spherical, inverted tear-shaped or pancake like. Also of interest is the finite time singularity which occurs when the drop pinches off; this is seen to be of the power law type with an exponent of 2/3.

Histograms are convenient and easy-to-use tools for estimating the shape of data, but they have serious problems which are magnified for multivariate data. We combine classic histograms with shape estimation by polynomials. The new relatives, `polynomial histograms', have surprisingly nice mathematical properties, which we will explore in this talk. We also show how they can be used for real data of 10-20 dimensions to analyse and understand the shape of these data.

Noncommutative geometry (in the sense of Alain Connes) involves replacing a conventional space by a "space" in which the algebra of functions is noncommutative. The simplest truly non-trivial noncommutative manifold is the noncommutative 2-torus, whose algebra of functions is also called the irrational rotation algebra. I will discuss a number of recent results on geometric analysis on the noncommutative torus, including the study of nonlinear noncommutative elliptic PDEs (such as the noncommutative harmonic map equation) and noncommutative complex analysis (with noncommutative elliptic functions).

Hirschsprung's disease is relatively common, affecting roughly 1 in 5000 newly born babies each year in Australia. The disease occurs when there is an incomplete formation of the nervous system in the gut. Mathematical models can help in determining the underlying mechanisms that cause the disease. Comparisons between theoretical predictions and experimental results will be made.

The Chern-Weil homomorphism gives a geometric method for calculating characteristic classes for principal bundles. In infinite dimensions, however, the standard theory fails due to analytical problems. In this talk I shall give a geometric method for calculating characteristic classes for principal bundle with structure group the loop group of a compact group which side-steps these complications. This theory is inspired in some sense by results on the string class (a certain cohomology class on the base of a loop group bundle) which I shall outline.

The human brain has evolved to be able to think intuitively in three dimensions. Unfortunately the real world is at least four and maybe 10, 11 or 26 dimensional. In this talk I will discuss some of the tricks mathematicians have devised to think about higher dimensional objects.

We introduce the four classes of complex manifolds defined by having few or many holomorphic maps to or from the complex plane. Two of these classes have played an important role in complex geometry for a long time. A third turns out to be too large to be of much interest. The fourth class has only recently emerged from work of Abel Prize winner Mikhail Gromov.

The study of wall-bounded turbulent flows has received great attention over the past few years as a result of high Reynolds number experiments conducted in new high Reynolds number facilities such as the Princeton "superpipe", the NDF facility in Chicago and the HRNBLWT at the University of Melbourne. These experiments have brought into question the fundamental scaling laws of the turbulence and mean flow quantities as well as revealed high Reynolds number phenomena, which make extrapolation of low Reynolds number results highly questionable. In this talk these issues will be reviewed and new results from the HRNBLWT and atmospheric surface layer on the salt-flats of Utah will be presented documenting unique high Reynolds number phenomena. The implications for skin-friction drag reduction technologies and improved near-wall models for large-eddy simulation will be discussed.

A compact K{\"a}hler manifold $X$ is a holomorphic symplectic manifold if it admits a non-degenerate holomorphic two-form $\sigma$. According to a theorem of Matsushita, fibrations on $X$ must be of a very restricted type: the fibres must be Lagrangian with respect to $\sigma$ and the generic fibre must be a complex torus. Moreover, it is expected that the base of the fibration must be complex projective space, and this has been proved by Hwang when $X$ is projective. The simplest example of these {\em Lagrangian fibrations\/} are elliptic K3 surfaces. In this talk we will explain the role of elliptic K3s in the classification of K3 surfaces, and the (conjectural) generalization to higher dimensions.

The loop space LM of a Riemannian manifold M comes with a family of Riemannian metrics indexed by a Sobolev parameter. We can construct characteristic classes for LM using the Wodzicki residue instead of the usual matrix trace. The Pontrjagin classes of LM vanish, but the secondary or Chern-Simons classes may be nonzero and may distinguish circle actions on M. There are similar results for diffeomorphism groups of manifolds.

The Hilbert scheme ${\mathrm Hilb}^nS$ of points on a K3 surface $S$ is a well-known holomorphic symplectic manifold. When does ${\mathrm Hilb}^nS$ admit a Lagrangian fibration? The existence of a Lagrangian fibration places some conditions on the Hodge structure, since the pull back of a hyperplane from the base gives a special divisor on ${\mathrm Hilb}^nS$, and in turn a special divisor on $S$. The converse is more difficult, but using Fourier-Mukai transforms we will show that if $S$ admits a divisor of a certain degree then ${\mathrm Hilb}^nS$ admits a Lagrangian fibration.

Given a certain kind of submanifold $Y$ of a symplectic manifold $(X,\omega)$ we can form its coisotropic reduction as follows. The null directions of $\omega|_Y$ define the characteristic foliation $F$ on $Y$. The space of leaves $Y/F$ then admits a symplectic form, descended from $\omega|_Y$. Locally, the coisotropic reduction $Y/F$ looks just like a symplectic quotient. This construction also work for holomorphic symplectic manifolds, though one of the main difficulties in practice is ensuring that the leaves of the foliation are compact. We will describe a criterion for compactness, and apply coisotropic reduction to produce a classification result for Lagrangian fibrations by Jacobians.

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.

Gaboriau and Popa showed that a non-abelian free group on finitely many generators has continuum many measure preserving, free, ergodic, actions on standard Borel probability spaces. The original proof used the notion of property (T). I will sketch how this can be replaced by an elementary, and apparently new, dynamical property.

The Stein-Tomas restriction theorem says that the Fourier transform of a function in L^p(R^n) restricts to an L^2 function on the unit sphere, for p in some range [1, 2(n+1)/(n+3)]. I will discuss geometric generalizations of this result, by interpreting it as a property of the spectral measure of the Laplace operator on R^n, and then generalizing to the Laplace-Beltrami operator on certain complete Riemannian manifolds. It turns out that dynamical properties of the geodesic flow play a crucial role in determining whether a restriction-type theorem holds for these manifolds.

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. In this talk, I will explain why turbulent flows are difficult to predict and describe a modern mathematical model of turbulence based on a random collection of fluid vortices.

A basic example of a Hopf algebra is a group algebra: it is the vector space having the group as basis and having multiplication linearly extending that of the group. We can start with a category instead of a group, form the free vector space on the set of its morphisms, and define multiplication to be composition when possible and zero when not. The multiplication has an identity if the category has finitely many objects; this is a basic example of a weak bialgebra. It is a weak Hopf algebra when the category is a groupoid. Group algebras are also Frobenius algebras. We shall generalize weak bialgebras and Frobenius algebras to the context of monoidal categories and describe some of their theory using the geometry of string diagrams.

The flows generated in many modern micro-devices possess very little convective inertia, however, they can be highly unsteady and exert substantial hydrodynamic forces on the device components. Typically these components exhibit some degree of compliance, which traditionally has been treated using simple one-dimensional elastic beam models. However, recent findings have suggested that three-dimensional effects can be important and, accordingly, we consider the elastohydrodynamic response of a rapidly oscillating three-dimensional elastic plate that is immersed in a viscous fluid. In addition, a preliminary model will be presented which incorporates the presence of a nearby elastic wall.

The ordinary integral on L^1 functions on R^d unfortunately does not extend to a translation invariant linear form on the whole algebra of pseudodifferential symbols on R^d, forcing to work with ordinary linear extensions which fail to be translation invariant. Defect formulae which express the difference between various linear extensions, show that they differ by local terms involving the noncommutative residue. In particular, we shall show how integrals regularised by a "dimensional regularisation" procedure familiar to physicists differ from Hadamard finite part (or "cut-off" regularised) integrals by a residue. When extended to pseudodifferential operators on closed manifolds, these defect formulae express the zeta regularised traces of a differential operator in terms of a residue of its logarithm. In particular, we shall express the index of a Dirac type operator on a closed manifold in terms of a logarithm of a generalized Laplacian, thus giving an a priori local description of the index and shall discuss further applications.

The simple groups are the building blocks of all finite groups. The classification of finite simple groups is a towering achievement of 20th century mathematics. In addition to 18 infinite families of finite simple groups, there are 26 sporadic groups. The biggest sporadic group, dubbed The Monster, has about 10^54 elements. The talk will give a glimpse of this deep area of mathematics.

Bundles of C*-algebras over a topological space M can be classified by a Dixmier-Douady obstruction in H^3(M,Z). This talk will describe some recent work with Mathai investigating the relationship between algebra bundles on M and on its covering space, where there can be no obstruction, particularly when there is a group acting on M.

Given a polynomial in two or more variables, one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called "tropical varieties." Mohammed Abouzaid received a B.S. in 2002 from the University of Richmond, and a Ph.D. in 2007 from the University of Chicago under the supervision of Paul Seidel. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology, in particular the homological mirror symmetry conjecture. Since 2007 he has been a postdoctoral fellow at MIT, and a Clay Mathematics Institute Research Fellow.

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.

Fourier-Mukai transforms are a geometric analogue of integral transforms playing an important role in algebraic geometry. Their name derives from the construction of Mukai involving the Poincare line bundle associated to an abelian variety. In this talk I will discuss recent work looking at an analogue of this original Fourier-Mukai transform in the context of differential geometry, which gives an isomorphism between the invariant differential cohomology of a real torus and its dual.

Berger's classification of irreducibly represented Lie groups that can occur as holonomy groups of semi-Riemannian manifolds is a remarkable result of modern differential geometry. What is remarkable about it is that it is so short and that only so few types of geometry can occur. In Riemannian signature this is even more remarkable, taking into account that any representation of a compact Lie group admits a positive definite invariant scalar product. Hence, for any not too small n there is an abundance of irreducible subgroups of SO(n). We show that in other signatures the situation is quite different with, for example, SO(1,n) having no proper irreducible subgroups. We will show how this and the corresponding result about irreducible subgroups of SO(2,n) follows from the Karpelevich-Mostov theorem. (This is joint work with Antonio J. Di Scala, Politecnico di Torino.)

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.

We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.

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

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

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

In joint work with Ajneet Dhillon, we find upper bounds for the essential dimension of various moduli stacks of SL_n-bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the moduli stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.

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

We will review holomorphic extension problems starting with the famous Hartogs extension theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent (partial) results of Al Boggess (Texas A&M Univ.), Zbigniew Slodkowski (Univ. Illinois at Chicago), and the speaker. The holomorphic extension problems for holomorphic or Cauchy-Riemann functions are fundamental problems in complex analysis of several variables. The talk will be very elementary, with many figures, and accessible to graduate and even advanced undergraduate students.

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

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

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 sequence first, second, third,... can be continued with infinite ordinal numbers. I will explain what these infinite numbers are and how they can be used -- and sometimes must be used! -- to prove facts about ordinary, finite numbers.

The n-sphere considered as a conformal manifold can be viewed as the projectivisation of the light cone in n+2 Minkowski space. A construction that generalises this picture to arbitrary conformal classes is the ambient metric introduced by C. Fefferman and R. Graham. In the talk, I will explain the Fefferman-Graham ambient metric construction and how it detects the existence of certain metrics in the conformal class. Then I will present conformal classes of signature (3,2) for which the 7-dimensional ambient metric has the noncompact exceptional Lie group G_2 as its holonomy. This is joint work with P. Nurowski, Warsaw University.

Jeffery–Hamel flows describe the steady two-dimensional flow of an incompressible viscous fluid between plane walls separated by an angle $\alpha$. They are often used to approximate the flow in domains of finite radial extent. However, whilst the base Jeffery–Hamel solution is characterised by a subcritical pitchfork bifurcation, studies in expanding channels of finite length typically find symmetry breaking via a supercritical bifurcation.

We use the finite element method to calculate solutions for flow in a two-dimensional wedge of finite length bounded by arcs of constant radii, $R_1$ and $R_2$. We present a comprehensive picture of the bifurcation structure and nonlinear states for a net radial outflow of fluid. We find a series of nested neutral curves in the Reynolds number-$\alpha$ plane corresponding to pitchfork bifurcations that break the midplane symmetry of the flow. We show that these finite domain bifurcations remain distinct from the similarity solution bifurcation even in the limit $R_2/R_1 \rightarrow \infty$.

We also discuss a class of stable steady solutions apparently related to a steady, spatially periodic, wave first observed by Tutty (1996). These solutions remain disconnected in our domain in the sense that they do not arise via a local bifurcation of the Stokes flow solution as the Reynolds number is increased.

This paper considers a model for asset pricing in a world where the randomness is modeled by a Markov chain rather than Brownian motion. In this paper we develop a theory of optimal stopping and related variational inequalities for American options in this model. A version of Saigal's Lemma is established and numerical algorithms developed. This is a joint work with John van der Hoek.

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.

Following Pearson in 1905, we study the expected distance of a two-dimensional walk in the plane with unit steps in random directions---what Pearson called a "ramble". A series evaluation and recursions are obtained making it possible to explicitly determine this distance for small number of steps. Closed form expressions for all the moments of a 2-step and a 3-step walk are given, and a formula is conjectured for the 4-step walk. Heavy use is made of the analytic continuation of the underlying integral.

In the mid-1990s the computer 'Deep Blue' beat Kasparov, the world chess champion. Will computers soon overtake us humans in other endeavours of intelligence? Roger Penrose's thesis is that human intelligence is far more subtle than has previously been imagined, that the quest for human-like artificial intelligence in computers, the holy grail of artificial intelligence, is hopeless. The argument ranges from icily clear mathematics of computation, through the amazing shadows of quantum physics, and thence to new conjectures in biology.

Suppose $G$ is a compact Lie group, $LG$ its (free) loop group and $\Omega G \subseteq LG$ its based loop group. Let $P \to M$ be a principal bundle with structure group one of these loop groups. In general, differential form representatives of characteristic classes for principal bundles can be easily obtained using the Chern-Weil homomorphism, however for infinite-dimensional bundles such as $P$ this runs into analytical problems and classes are more difficult to construct. In this talk I will explain some new results on characteristic classes for loop group bundles which demonstrate how to construct certain classes---which we call string classes---for such bundles. These are obtained by making heavy use of a certain $G$-bundle associated to any loop group bundle (which allows us to avoid the problems of dealing with infinite-dimensional bundles). We shall see that the free loop group case naturally involves equivariant cohomology.

In this talk, I shall briefly review the notion of stability for holomorphic vector bundles on compact complex manifolds as discussed in the first part of this talk (28 August 2009). Then I shall attempt to compute some explicit examples in simple situations, illustrating the use of basic algebraic-geometric tools. The level of the talk will be appropriate for graduate students, particularly those who have been taking part in the algebraic geometry reading group meetings.

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

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

The caloron transform is a `fake' dimensional reduction which transforms a G-bundle over certain manifolds to a loop group of G bundle over a manifold of one lower dimension. This talk will review the caloron transform and show how it can be best understood using the language of pseudo-isomorphisms from category theory as well as considering its application to Bogomolny monopoles and string structures.

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.

This talk is a continuation of the talk on 30 April. The same abstract applies: In this talk, I shall briefly review the notion of stability for holomorphic vector bundles on compact complex manifolds as discussed in the first part of this talk (28 August 2009). Then I shall attempt to compute some explicit examples in simple situations, illustrating the use of basic algebraic-geometric tools. The level of the talk will be appropriate for graduate students, particularly those who have been taking part in the algebraic geometry reading group meetings.

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

High on a mathematician's to-do list is classifying objects and structures that arise in mathematics. We see patterns in things and want to know what other sorts of things behave similarly. This poses several problems. How can you tell when two seemingly different mathematical objects are the same? Can you even tell when two seemingly similar mathematical objects are the same? In fact, what does "the same" even mean? How can you tell if two things are the same when you can't even see them! In this talk, we will take a walk through some areas of maths known as algebraic topology and category theory and I will show you some of the ways mathematicians have devised to tell when two things are "the same".

The Whitehead tower of a topological space seeks to resolve that space by successively removing homotopy groups from the 'bottom up'. For a path-connected space with no 1-dimensional local pathologies the first stage in the tower can be chosen to be the universal (=1-connected) covering space. This construction also works in the category Diff of manifolds. However, further stages in the two known constructions of the Whitehead tower do not work in Diff, being purely topological - and one of these is non-functorial, depending on a large number of choices. This talk will survey results from my thesis which constructs a new, functorial model for the 2-connected cover which will lift to a generalised (2-)category of smooth objects. This talk contains joint work with Andrew Stacey of the Norwegian University of Science and Technology.

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

We show uniqueness up to sign of positive, orthogonal almost-Kahler structures on any non-scalar flat Kahler-Einstein surface. This is joint work with A. J. di Scala.

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.

A basic operation in calculus of variations is the Euler-Lagrange variational derivative, whose kernel determines the extremals of functionals. There exists a natural resolution of this operator, called the variational complex. In this talk, I shall explain how to use tools from the theory of vertex algebras to explicitly construct the variational complex. This also provides a very convenient language for classifying and constructing integrable Hamiltonian evolution equations.

Last time I introduced the variational complex of an algebra of differential functions and gave a sketchy definition of a vertex algebra. This week I will make this notion more precise and explain how to apply it to the calculus of variations.

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

Research into two-dimensional laminar fluid mixing has enjoyed a renaissance in the last decade since the realisation that the Thurston–Nielsen theory of surface homeomorphisms can assist in designing efficient "topologically chaotic" batch mixers. In this talk I will survey some tools used in topological fluid kinematics, including braid groups, train-tracks, dynamical systems and topological index formulae. I will then make some speculations about topological chaos in three dimensions.

I will describe a family of algebras of tangles (which give rise to link invariants following the methods of Reshetikhin-Turaev and Jones) and describe some aspects of their structure and their representation theory. The main goal will be to explain how to use universal Verma modules for the symplectic group to compute the representation theory of affine BMW (Birman-Murakami-Wenzl) algebras.

I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)

I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)

I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)

I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)

I shall give an overview of recent progress in homological mirror symmetry, both in clarifying our conceptual understanding of how the sign of the canonical bundle affects the behaviour of the mirror, and in obtaining concrete examples where the mirror conjecture has now been verified. (This is a two-hour talk.)

Quillen introduced a $K'_0$-theory for possibly nonunital rings and showed that it agrees with the usual algebraic $K_0$-theory if the ring is unital. We shall introduce higher $K'$-groups for $k$-algebras, where $k$ is a field, and discuss some elementary properties of this theory. We shall also show that for stable $C*$-algebras the higher $K'$-theory agrees with the topological $K$-theory. If time permits we shall explain how this provides a formalism to treat topological $\mathbb{T}$-dualities via Kasparov's bivariant $K$-theory.

The Mathematica Seminars 2010 offer an opportunity to experience the applicability, ease-of-use, as well as the advancements of Mathematica 7 in education and academic research. These seminars will highlight the latest directions in technical computing with Mathematica, and the impact this technology has across a wide range of academic fields, from maths, physics and biology to finance, economics and business. Those not yet familiar with Mathematica will gain an overview of the system and discover the breadth of applications it can address, while experts will get firsthand experience with recent advances in Mathematica like parallel computing, digital image processing, point-and-click palettes, built-in curated data, as well as courseware examples.

As a tool for studying enumerative problems in geometry Eynard and Orantin associate multilinear differentials to any plane curve. Their work comes from matrix models but does not require matrix models (for understanding or calculations). In some sense they describe deformations of complex structures of a curve and conjectural relationships to deformations of Kahler structures of an associated object. I will give an introduction to their invariants via explicit examples, mainly to do with the moduli space of Riemann surfaces, in which the plane curve has genus zero.

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.

A point process rainfall model is further developed that has storm origins occurring in space-time according to a Poisson process. Each storm origin has a random radius so that storms occur as circular regions in two-dimensional space, where the storm radii are taken to be independent exponential random variables. Storm origins are of random type z, where z follows a continuous probability distribution. Cell origins occur in a further spatial Poisson process and have arrival times that follow a Neyman-Scott point process. Cell origins have random radii so that cells form discs in two-dimensional space. Statistical properties up to third order are derived and used to fit the model to 10 min series taken from 23 sites across the Roma region, Italy. Distributional properties of the observed annual maxima are compared to equivalent values sampled from series that are simulated using the fitted model. The results indicate that the model will be of use in urban drainage projects for the Roma region.

The aim of the talk is to give an overview of the noncommutative geometry approach to index theory.

Simplicial sets became popular in the 1950s as a combinatorial way to study the homotopy theory of topological spaces. They are more robust than the older notion of simplicial complexes, which were introduced for the same purpose. In this talk, which will be as introductory as possible, we will review some classical functors arising in the theory of simplicial sets, some well-known, some not-so-well-known. We will re-examine the proof of an old theorem of Kan in light of these functors. We will try to keep all jargon to a minimum.

In linear regression analysis, randomness often exists in the independent variables and the resulting models are referred to errors-in-variables (EIV) models. The existing general EIV modeling framework, the structural model approach, is parametric and dependent on the usually unknown underlying distributions. In this work, we introduce a general non-parametric EIV modeling framework, the compound regression analysis, featuring an intuitive geometric representation and a 1-1 correspondence to the structural model. Properties, examples and further generalizations of this new modeling approach are discussed in this talk.

In my previous talk I introduced a functor from the category of k-algebras (k field) to abelian groups, called KQ-theory. In this talk I will explain its relationship with topological (homological) T-dualities and twisted K-theory.

Euclidean space is flat but the real world is curved. This causes lots of problems for sailors, surveyors, mapmakers, and even geometers. In the talk I will explain how the notion of curvature evolved in mathematics starting off from practical applications such as geodesy and cartography and yielding less practical applications in modern physics.

Generalised varying-coefficient models (GVC) are very important models. There are a considerable number of literature addressing these models. However, most of the existing literature are devoted to the estimation procedure. In this talk, I will systematically investigate the statistical inference for GVC, which includes confidence band as well as hypothesis test. I will show the asymptotic distribution of the maximum discrepancy between the estimated functional coefficient and the true functional coefficient. I will compare different approaches for the construction of confidence band and hypothesis test. Finally, the proposed statistical inference methods are used to analyse the data from China about contraceptive use there, which leads to some interesting findings.

For each automorphism, $\alpha$, of the locally compact group $G$ there is a corresponding {\sl contraction subgroup\/}, $\hbox{con}(\alpha)$, which is the set of $x\in G$ such that $\alpha^n(x)$ converges to the identity as $n\to \infty$. Contractions subgroups are important in representation theory, through the Mautner phenomenon, and in the study of convolution semigroups. If $G$ is a Lie group, then $\hbox{con}(\alpha)$ is automatically closed, can be described in terms of eigenvalues of $\hbox{ad}(\alpha)$, and is nilpotent. Since any connected group may be approximated by Lie groups, contraction subgroups of connected groups are thus well understood. Following a general introduction, the talk will focus on contraction subgroups of totally disconnected groups. A criterion for non-triviality of $\hbox{con}(\alpha)$ will be described (joint work with U.~Baumgartner) and a structure theorem for $\hbox{con}(\alpha)$ when it is closed will be presented (joint with H.~Gl\"oeckner).

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.

The sense of smell is important in nature, but the least well understood of our senses. A mathematical model of smell, which combines the transmission of volatile-organic-compound chemical signals (VOCs) on the wind, transduced by olfactory receptors in our noses into neural information, and assembled into our odour perception, is useful. Applications include regulations for odour nuisance, like German VDI protocols for calibrated noses, to the design of modern chemical sensors for extracting information from the environment and even for the perfume industry. This talk gives a broad overview of turbulent mixing in surface layers of the atmosphere, measurements of VOCs with PTR-MS (proton transfer reaction mass spectrometers), our noses, and integrated environmental models of the Alumina industry (a source of odour emissions) to help understand the science of smell.

In the process of determining, amongst other things, the optimal strategy for playing a game show, and explaining the apparent persistence of a population that can be shown to die out with certainty, we will encounter a car, numerous goats, at least four doors, a frog, four lily pads, and some applied probability.

Classical gauge theories study sections of vector bundles and associated connections and curvature. The corresponding quantum gauge theories are normally written algebraically but can be understood as noncommutative geometries. This talk will describe one approach to the quantum gauge theories which uses braided categories.

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/

Animals typically search for food and mates, while avoiding predators. This is particularly critical for keystone organisms such as intertidal gastropods and copepods (i.e. millimeter-scale crustaceans) as they typically rely on non-visual senses for detecting, identifying and locating mates in their two- and three-dimensional environments. Here, using stochastic methods derived from the field of nonlinear physics, we provide new insights into the nature (i.e. innate vs. acquired) of the motion behavior of gastropods and copepods, and demonstrate how changes in their behavioral properties can be used to identify the trade-offs between foraging for food or sex. The gastropod Littorina littorea hence moves according to fractional Brownian motions while foraging for food (in accordance with the fractal nature of food distributions), and switch to Brownian motion while foraging for sex. In contrast, the swimming behavior of the copepod Temora longicornis belongs to the class of multifractal random walks (MRW; i.e. a form of anomalous diffusion), characterized by a nonlinear moment scaling function for distance versus time. This clearly differs from the traditional Brownian and fractional Brownian walks expected or previously detected in animal behaviors. The divergence between MRW and Levy flight and walk is also discussed, and it is shown how copepod anomalous diffusion is enhanced by the presence and concentration of conspecific water-borne signals, and is dramatically increasing male-female encounter rates.

This paper builds on earlier work by Davis and Hobson (Mathematical Finance, 2007) giving model-free---except for a 'frictionless markets' assumption--- necessary and sufficient conditions for absence of arbitrage given a set of current-time put and call options on some underlying asset. Here we suppose that the prices of a set of put options, all maturing at the same time, are given and satisfy the conditions for consistency with absence of arbitrage. We now add a path-dependent option, specifically a weighted variance swap, to the set of traded assets and ask what are the conditions on its time-0 price under which consistency with absence of arbitrage is maintained. In the present work, we work under the extra modelling assumption that the underlying asset price process has continuous paths. In general, we find that there is always a non- trivial lower bound to the range of arbitrage-free prices, but only in the case of a corridor swap do we obtain a finite upper bound. In the case of, say, the vanilla variance swap, a finite upper bound exists when there are additional traded European options which constrain the left wing of the volatility surface in appropriate ways.

In part I of this talk (JC seminar May 2008) we presented motivation and the basic definitions for building homotopy theory into an arbitrary category by introducing the notion of (higher) stacks. In part II we consider a specific example on the category of schemes to illustrate how the machinery works in practice. It will lead us into motivic territory (if we like it or not).

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that -\partial^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.

We consider a system where jobs of several types are served by servers of several types, and a bipartite graph between server types and job types describes feasible assignments. This is a common situation in manufacturing, call centers with skill based routing, matching of parent-child in adoption or matching in kidney transplants etc. We consider the case of first come first served policy: jobs are assigned to the first available feasible server in order of their arrivals. We consider two types of policies for assigning customers to idle servers - a random assignment and assignment to the longest idle server (ALIS) We survey some results for four different situations:

• For a loss system we find conditions for reversibility and insensitivity.
• For a manufacturing type system, in which there is enough capacity to serve all jobs, we discuss a product form solution and waiting times.
• For an infinite matching model in which an infinite sequence of customers of IID types, and infinite sequence of servers of IID types are matched according to first come first, we obtain a product form stationary distribution for this system, which we use to calculate matching rates.
• For a call center model with overload and abandonments we make some plausible observations.

This talk surveys joint work with Ivo Adan, Rene Caldentey, Cor Hurkens, Ed Kaplan and Damon Wischik, as well as work by Jeremy Visschers, Rishy Talreja and Ward Whitt.

Horizontal convecton refers to flows driven by uneven heating on a horizontal forcing boundary. Flows exhibiting these characteristics are prevalent in nature, and include the North-South Hadley circulation within the atmosphere between warmer and more temperate latitudes, as well as ocean currents driven by non-uniform heating via solar radiation.

Here a model for these generic convection flows is established featuring a rectangular enclosure, insulated on the side and top walls, and driven by a linear temperature gradient applied along the bottom wall. Rayleigh number dependence of heat transfer through the forcing boundary is computed and compared with theory. Attention is given to transitions in the flow, including the development of unsteady flow and three-dimensional flow: the effect of these transitions on the Nusselt-Rayleigh number scaling exponents is described.

The p-adic numbers are: (a) something that visiting seminar speakers invoke when the want to frighten the audience; (b) a fascinating and useful concept in modern algebra; (c) alphabetically just before q-adic numbers? In this talk I hope to convince the audience that option (b) is worth considering. I will begin by reviewing how we get from integers via rational numbers to the real number system. Then we'll look at how this process can be "twisted" to produce something new.

After a quick introduction to real and complex analytic sets, I will discuss possible notions of complex dimension of real sets, and then discuss a structure theorem for the holomorphic closure dimension which is defined as the dimension of the smallest complex analytic germ containing the real germ.

Have you ever wondered where the weather forecast for the next seven days comes from? Come and find out! We will look at the basic laws of meteorology, leading in to the primitive equations, which are solved on supercomputers to produce the weather forecasts we see every day. We will finish by using the current numerical weather prediction charts to forecast our weather for the next few days.

Given a real analytic set R, denote by A the subset of R of points through which there is a nontrivial complex variety contained in R, i.e., A consists of points in R of positive complex dimension. I will discuss the structure of the set A.

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

We show that there are only three regular tilings of the plane, that is, tilings using a regular polygon tile, with tile vertices touching. We also classify the semiregular tilings; tilings using more than one type of regular polygon. These tilings all have many symmetries, in particular, we can translate the tiling, and it still looks the same. Sir Roger Penrose constructed a set of aperiodic tiles; a tiling using these Penrose tiles has no translational symmetry, that is, a translated copy will never match the original. We look at some of the interesting properties of these tiles.

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

Black and Scholes have introduced a new approach to model the stock price dynamics about three decades ago. The so called Black Scholes model seems to be very adapted to the nature of market prices, mainly because the usage of the Brownian motion and the mathematical properties that follow from. Like every theoretical model, put in practice, it does not appear to be flawless, that means that new adaptations and extensions should be made so that engineers and marketers could utilise the Black Scholes models to trade and hedge risk on the market. A more detailed description with application will be given in the talk.

I hope we don't have to wait until oil and coal run out before we tackle that." - Thomas Edison, 1931. In a bid to resolve energy issues consistent with Thomas Edison's worries, scientists have been looking at other clean and sustainable sources of energy such as natural gas - methane. In this talk, the interaction between a methane molecule and carbon nanotubes is investigated mathematically, using two different models - first discrete and second, continuous. These models are analyzed to determine the dimensions of the particular nanotubes which will readily suck-up methane molecules. The results determine the minimum and maximum interaction energies required for methane encapsulation in different tube sizes, and establish the second model of the methane molecule as a simple and elegant model which might be exploited for other problems.

A parallel lightlike vector field on a Lorentzian manifold X naturally defines a foliation of codimension 1 on X and a 1-dimensional subfoliation. In the first part we introduce Lorentzian metrics on the total space of certain circle bundles in order to construct weakly irreducible Lorentzian manifolds admitting a parallel lightlike vector field such that all leaves of the foliations are compact. Then we study which holonomy representations can be realized in this way. Finally, we consider the structure of arbitrary Lorentzian manifolds for which the leaves of the foliations are compact.

Have you thought about what sort of car you would be driving in the future? Would it be a hybrid, solar, hydrogen or electric car? I would like to be driving a hydrogen car because my field of research may aid in their development! In my presentation I will introduce you to the world of metal-organic frameworks, which are an exciting new class of materials that have great potential in applications such as hydrogen gas storage. I will also discuss about the mathematical model that I am using to model the performance of metal-organic frameworks based on beryllium.

Woronowicz initiated the study of quantum groups using C*-algebras. His framework enabled him to deal with compact (linear) quantum groups. In this talk we shall introduce a notion of quantum groups that can handle infinite dimensional examples like SU(\infty). We shall also study some quantum homogeneous spaces associated to this group and compute their K-theory groups. This is joint work with V. Mathai.

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

The lakes and weirs along the lower Murray River in Australia are aggregated and considered as a sequence of five reservoirs. A seasonal Markov chain model for the system will be implemented, and a stochastic dynamic program will be used to find optimal release strategies, in terms of expected monetary value (EMV), for the competing demands on the water resource given the stochastic nature of inflows. Matrix analytic methods will be used to analyse the system further, and in particular enable the full distribution of first passage times between any groups of states to be calculated. The full distribution of first passage times can be used to provide a measure of the risk associated with optimum EMV strategies, such as conditional value at risk (CVaR). The sensitivity of the model, and risk, to changing rainfall scenarios will be investigated. The effect of decreasing the level of discretisation of the reservoirs will be explored. Also, the use of matrix analytic methods facilitates the use of hidden states to allow for hydrological persistence in the inflows. Evidence for hydrological persistence of inflows to the lower Murray system, and the effect of making allowance for this, will be discussed.

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.

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

A key question in mathematical finance is: given a future random payoff X, what is its value today? If X represents a loss, one can ask how risky is X. To mitigate risk it must be modelled and quantified. The finance industry has used Value-at-Risk and conditional Value-at-Risk as measures. However, these measures are not time consistent and Value-at-Risk can penalize diversification. A modern theory of risk measures is being developed which is related to solutions of backward stochastic differential equations in continuous time and stochastic difference equations in discrete time. I first review risk measures used in mathematical finance, including static and dynamic risk measures. I recall results relating to backward stochastic difference equations (BSDEs) associated with a single jump process. Then I evaluate some numerical examples of the solutions of the backward stochastic difference equations and related risk measures. These concepts are new. I hope the examples will indicate how they might be used.

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

Renewable energy has been one of the main issues nowadays. The implications of fossil energy and nuclear energy along with its limited source have triggered researchers and industries to find another source of renewable energy for example hydro energy, wind energy and also wave energy. In this seminar, I will talk about the spectral estimation and wavelet estimation of a linear dynamical system of motion for a heaving buoy wave energy device. The spectral estimates was based on the Fourier transform, while the wavelet estimate was based on the wavelet transform. Comparisons between two spectral estimates with a wavelet estimate of the amplitude response operator(ARO) for the dynamical system of the wave energy device shows that the wavelet estimate ARO is much better for data with and without noise.

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.

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the populations size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Here we will discuss the use of diffusion approximations to perform estimation in several scenarios, with successively reduced assumptions: (i) under the assumption of stationarity (the process had been evolving for a very long time with constant parameter values); (ii) transient dynamics (the assumption of stationarity is invalid, and thus only constant parameter values may be assumed); and, (iii) time-inhomogeneous chains (the parameters may vary with time) and accounting for observation error (a sample of the true state is observed).

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.

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

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.

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

The Euler characteristic is a nice simple integer invariant that one can attach to a space. Unfortunately, it is not natural: maps between spaces do not induce maps between their Euler characteristics, because it makes no sense to talk of a map between integers. This shortcoming is fixed by homology. Maps between spaces induce maps between their homologies, with the Euler characteristic encoded inside the homology. Recently it has become possible to play the same game with knots and the Jones polynomial: the Khovanov homology of a knot both encodes the Jones polynomial and is a natural invariant of the knot. After saying what all this means, this talk will observe that Khovanov homology is just a special case of sheaf homology on a poset, and we will explore some of the ramifications of this observation. This is joint work with Paul Turner (Geneva/Fribourg).

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.

I will review what it means to lift the structure group of a principal bundle and the topological obstruction to this in the case of a central extension. I will then discuss some new results in the case of abelian extensions.

Permeability is a key parameter important to a variety of applications in geological engineering and in the environmental geosciences. The conventional definition of Darcy flow enables the estimation of permeability at different levels of detail. This lecture will focus on the measurement of surface permeability characteristics of a large cuboidal block of Indiana Limestone, using a surface permeameter. The paper discusses the theoretical developments, the solution of the resulting triple integral equations and associated computational treatments that enable the mapping of the near surface permeability of the cuboidal region. This data combined with a kriging procedure is used to develop results for the permeability distribution at the interior of the cuboidal region. Upon verification of the absence of dominant pathways for fluid flow through the cuboidal region, estimates are obtained for the "Effective Permeability" of the cuboid using estimates proposed by Wiener, Landau and Lifschitz, King, Matheron, Journel et al., Dagan and others. The results of these estimates are compared with the geometric mean, derived form the computational estimates.

This week, my parents will join around 185,000 other Australians, in a significant move towards renewable energy, and install solar panels on the roof of their house. While solar energy is an important and useful form of renewable energy it is not able to provide power all the time. Opponents of renewable energy maintain that until renewable energy can provide energy all the time, traditional fossil-fuel generated power will be required to produce our base-load power. Geothermal energy is a renewable energy that can provide energy all the time. However, due to its special geological requirements, it can only be produced in a very small number of places in the world. An Enhanced Geothermal System (EGS) is a new technology which allows geothermal energy to be produced in a much wider range of places than traditional geothermal energy. Currently, there are ten different companies investigating possible EGS sties within Australia. This seminar investigates the question, that all these companies hope they have answered well, 'Where is the best place in Australia for an EGS facility?'

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

In this talk I will review existing analytical results for priority queueing systems with Poisson incoming flows, general service times and a single server which needs some (random) time to switch between requests of different priority. Specifically, I will discuss analytical results for the busy period and workload of such systems with a special structure of switchover times. The results related to the busy period can be seen as generalisations of the famous Kendall-Tak\'{a}cs functional equation for $M|G|1$: being formulated in terms of Laplace-Stieltjes transform, they represent systems of functional recurrent equations. I will present a methodology and algorithms of their numerical solution; the efficiency of these algorithms is achieved by acceleration of the numerical procedure of solving the classical Kendall-Tak\'{a}cs equation. At the end I will identify open problems with regard to such systems; these open problems are mainly related to the modelling of switchover times.

The Hochschild cochain complex of an associative algebra provides an important bridge between algebra and geometry. Algebraically, this is the derived center of the algebra. Geometrically, the Hochschild cohomology of the algebra of smooth functions on a manifold is isomorphic to the graduate space of polyvector fields on this manifold. There are many important operations acting on the Hochschild complex. It is, however, a tricky question to ask which operations are natural because the Hochschild complex is not a functor. In my talk I will explain how we can overcome this obstacle and compute all possible natural operations on the Hochschild complex. The result leads immediately to a proof of the Deligne conjecture on Hochschild cochains.

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.

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.

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.

The field of Object Oriented Data Analysis has made a lot of progress on the statistical analysis of the variation in populations of complex objects. A particularly challenging example of this type is populations of tree-structured objects. Deep challenges arise, which involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. These challenges, together with three completely different approaches to addressing them, are illustrated using a real data example, where each data point is the tree of blood arteries in one person's brain.

Tensors are important objects that are frequently used in a variety of fields including continuum mechanics, general relativity and differential geometry. Despite their importance, they are often defined poorly (if at all) which contributes to a lack of understanding. In this talk, I will give a concrete definition of a tensor and provide some familiar examples. For the remainder of the talk, I will discuss some applications—here I mean applications in the pure maths sense (i.e. more abstract nonsense, but hopefully still interesting).

I will explain how to construct a bivariant Chern-Connes character on the category of sigma-C*-algebras taking values in Puschnigg's local cyclic homology. Roughly, setting the first (resp. the second) variable to complex numbers one obtains the K-theoretic (resp. dual K-homological) Chern-Connes character in one variable. We shall focus on the dual K-homological Chern-Connes character and investigate it in the example of SU(infty).

The core of the Internet is made up of many different computers (called routers) in many different interconnected networks, owned and operated by many different organisations. A popular and important field of study in the past has been "network topology": for instance, understanding which routers are connected to which other routers, or which networks are connected to which other networks; that is, studying and modelling the connection structure of the Internet. Previous study in this area has been plagued by unreliable or flawed experimental data and debate over appropriate models to use. The Internet Topology Zoo is a new source of network data created from the information that network operators make public. In order to better understand this body of network information we would like the ability to randomly generate network topologies resembling those in the zoo. Leveraging previous wisdom on networks produced as a result of optimisation processes, we propose a simple objective function based on possible economic constraints. By changing the relative costs in the objective function we can change the form of the resulting networks, and we compare these optimised networks to a variety of networks found in the Internet Topology Zoo.

Let x be a real number. This familiar and seemingly innocent assumption opens up a world of infinite variety and information. We use some simple techniques (powers of two, geometric series) to examine some interesting consequences of generating random real numbers, and encounter both the best flash drive and the worst flash drive you will ever meet. Come "hold infinity in the palm of your hand", and contemplate eternity for about half an hour. Almost nothing is assumed, almost everything is explained, and absolutely all are welcome.

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.

Proteomic time-of-flight mass spectrometry produces a spectrum based on the peptides (chains of amino acids) in each patient’s serum sample. The spectra contain data points for an x-axis (peptide weight) and a y-axis (peptide frequency/count/intensity). It is our end goal to differentiate cancer (and sub-types) and control patients using these spectra. Before we can do this, peaks in these data must be found and common peptides to different spectra must be found. The data are noisy because of biotechnological variation and calibration error; data points for different peptide weights may in fact be same peptide. An algorithm needs to be employed to find common peptides between spectra, as performing alignment ‘by hand’ is almost infeasible. We borrow methods suggested in the literature by metabolomic gas chromatography-mass spectrometry and extend the methods for our purposes. In this talk I will go over the basic tenets of what we hope to achieve and the process towards this.

The distribution of individual orbits of unipotent flows in homogeneous spaces are well understood thanks to the work work of Marina Ratner. It is conjectured that this property is preserved on restricting the times from the integers to primes, this being important in the study of prime numbers as well as in such dynamics. We review progress in understanding this conjecture, starting with Dirichlet (a finite system), Vinogradov (rotation of a circle or torus), Green and Tao (translation on a nilmanifold) and Ubis and Sarnak (horocycle flows in the semisimple case).

For steady flows, the boundaries of Lagrangian Coherent Structures are segments of manifolds connected to fixed points. In the general unsteady situation, these boundaries are time-varying manifolds of hyperbolic trajectories. Locating these boundaries, and attempting to meaningfully quantify fluid flux across them, is difficult since they are moving with time. This talk uses a newly developed tangential movement theory to locate these boundaries in nearly-steady compressible flows.

Everything you have ever wanted to know about K3 surfaces! Two talks: 1:10 pm to 3:00 pm.

Textbooks remain a central part of mathematics lessons in secondary schools. However, while textbooks are still formatted in the traditional way, they are including increasingly more sophisticated software packages to assist teachers and students. I will be demonstrating the different software packages available to students included with two South Australian textbooks. I will talk about how these new features fit into the current classroom environment and some of their potential positives and negatives. I would also like to encourage people to share their own experiences with textbooks, especially if they were used in a novel way or you have experience of mathematics classes in another country.

According to a conjecture from the 90's, globally regular, static, spherically symmetric (i.e. particle-like) solutions with nonzero total magnetic charge are not expected to exist in Einstein-Yang-Mills theory. In this talk, I will describe recent work done in collaboration with M. Fisher where we establish the validity of this conjecture under certain restrictions on the residual gauge group. Of particular interest is that our non-existence results apply to the most widely studied models with Abelian residual groups.

The umbilical cord is the connecting cord between the developing embryo or fetus and the placenta. In a normal pregnancy it facilitates the supply of oxygen and nutrients from the placenta, in addition to the return of deoxygenated blood from the fetus. One of the most striking features of the umbilical cord is it's coiled structure, which allows the vasculature to withstand tensile and compressive forces in utero. The level of coiling also has a significant effect on the blood flow and cords exhibiting abnormally high or low levels are known to correlate well with adverse outcomes in pregancy, including fetal demise. In this talk I will discuss the complexities associated with numerically modeling blood flow within the umbilical cord, and provide an outline of the key features which will be investigated throughout my research.

We discuss the behaviour of the Oka property with respect to deformations of compact complex manifolds. We have recently proved that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G_delta subset of the base. We have also found a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. The special case when the fibres are tori will be considered, as well as the general case of holomorphic submersions with noncompact fibres.

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

Oka manifolds can be viewed as the "opposite" of Kobayashi hyperbolic manifolds. Kobayashi conjectured that the complement of a generic algebraic hypersurface of sufficiently high degree is hyperbolic. Therefore it is natural to ask whether the complement is Oka for the case of low degree or non-algebraic hypersurfaces. We provide a complete answer to this question for complements of hyperplane arrangements, and some results for graphs of meromorphic functions.

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.

Morava's extraordinary K-theories K(n) are a family of generalized cohomology theories which behave in some ways like K-theory (indeed, K(1) is mod 2 K-theory). Their construction exploits Quillen's description of cobordism in terms of formal group laws and Lubin-Tate's methods in class field theory for constructing abelian extensions of number fields. Constructed from homotopy-theoretic methods, they do not admit a geometric description (like deRham cohomology, K-theory, or cobordism), but are nonetheless subtle, computable invariants of topological spaces. In this talk, I will give an introduction to these theories, and explain how it is possible to define an analogue of twisted K-theory in this setting. Traditionally, K-theory is twisted by a three-dimensional cohomology class; in this case, K(n) admits twists by (n+2)-dimensional classes. This work is joint with Hisham Sati.

Just how many lobsters are there hanging around the South Australian coastline? How is this number changing over time? What is the demographic breakdown of this number? And what does it matter? Find out the answers to these questions in my upcoming talk. I will provide a brief flavour of the kinds of quantitative methods involved, showcasing relevant applications of regression, population modelling, estimation, as well as simulation. A product of these analyses are biological performance indicators which are used by government to help decide on fishery controls such as yearly total allowable catch quotas. This assists in maintaining the sustainability of the fishery and hence benefits both the fishers and the lobsters they catch.

Higher-rank graphs and their $C^*$-algebras were introduced by Kumjian and Pask in 2000. They have provided a rich source of tractable examples of $C^*$-algebras, the most elementary of which are the commutative algebras $C(\mathbb{T}^k)$ of continuous functions on $k$-tori. In this talk we shall describe how to define the homology and cohomology of a higher-rank graph, and how to associate to each higher-rank graph $\Lambda$ and $\mathbb{T}$-valued cocycle on $\Lambda$ a twisted higher-rank graph $C^*$-algebra. As elementary examples, we obtain all noncommutative tori. This is a preleminary report on ongoing joint work with Alex Kumjian and David Pask.

In the last two decades heap bioleaching has become established as a successful commercial option for recovering copper from low-grade secondary sulfide ores. Genetics-based approaches have recently been employed in the task of characterizing mineral processing bacteria. Data analysis is a key issue and thus the implementation of adequate mathematical and statistical tools is of fundamental importance to draw reliable conclusions. In this talk I will give a recount of two specific problems that we have been working on. The first regarding experimental design and the latter on modeling composition and activity of the microbial consortium.

In the finance or credit risk area, a bank may want to assess whether a client is going to default, or be able to meet the repayments. In the assessment of benign or malignant tumours, a correct diagnosis is required. In these and similar examples, we make decisions based on data. The classical t-tests provide a tool for making such decisions. However, many modern data sets have more variables than observations, and the classical rules may not be any better than random guessing. We consider Fisher's rule for classifying data into two groups, and show that it can break down for high-dimensional data. We then look at ways of overcoming some of the weaknesses of the classical rules, and I show how these "post-modern" rules perform in practice.

In physics T-duality is a phenomenon which relates certain types of string theories to one another. From a topological point of view, one can view string theory as a duality between line bundles carrying a degree three cohomology class (the H-flux). In this talk we will use bundle gerbes to give a geometric realisation of the H-flux and explain how to construct the T-dual of a line bundle together with its T-dual bundle gerbe.

Estimating disease prevalence in "hidden" populations such as injecting drug users or men who have sex with men is an important public health issue. However, traditional design-based estimation methods are inappropriate because they assume that a list of all members of the population is available from which to select a sample. Respondent Driven Sampling (RDS) is a method developed over the last 15 years for sampling from hidden populations. Similarly to snowball sampling, it leverages the fact that members of hidden populations are often socially connected to one another. Although RDS is now used around the world, there are several common population characteristics which are known to cause estimates calculated from such samples to be significantly biased. In this talk I'll discuss the motivation for RDS, as well as some of the recent developments in methods of estimation.

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

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

For many years industry has used CNC (computer numerically controlled) machines to craft specialist items. CNC machines traditionally mill out metal objects with arbitrary shapes, but they are expensive, large and dangerous. In recent years a new type of CNC machine has appeared - a 3D printer - which makes 3D objects by printing layers of plastic. These can be made safe, cheap, and small enough to fit on a desktop. I will show off my 3D printer, and explain some of the maths that goes into it.

In this talk we consider some statistical analyses of data arising in bioinformatics. The problems include the detection of differential expression in microarray gene-expression data, the clustering of time-course gene-expression data and, lastly, the analysis of modern-day cytometric data. Extensions are considered to the procedures proposed for these three problems in McLachlan et al. (Bioinformatics, 2006), Ng et al. (Bioinformatics, 2006), and Pyne et al. (PNAS, 2009), respectively. The latter references are available at http://www.maths.uq.edu.au/~gjm/.

In physics T-duality is a phenomenon which relates certain types of string theories to one another. From a topological point of view, one can view string theory as a duality between line bundles carrying a degree three cohomology class (the H-flux). In this talk we will use bundle gerbes to give a geometric realisation of the H-flux and explain how to construct the T-dual of a line bundle together with its T-dual bundle gerbe.

The Dirac operator was introduced by Paul Dirac in 1928 as the formal square root of the D'Alembert operator. Thirty years later it was rediscovered in Euclidean signature by Atiyah and Singer in their seminal work on index theory. In this talk I will describe efforts to construct a Dirac type operator on the classifying space for odd complex K-theory. Ultimately the aim is to produce a projective family of Fredholm operators realising elements in twisted K-theory of a certain moduli stack.

The study of molecular motion, interaction and space at the nanoscale has become a powerful tool in the area of gas separation, storage and conversion for efficient energy solutions. Modeling in this field has typically involved highly iterative computational algorithms such as molecular dynamics, Monte Carlo and quantum mechanics. Mathematical formulae in the form of analytical solutions to this field offer a range of useful and insightful advantages including optimization, bifurcation analysis and standardization. Here we present a few case scenarios where mathematics has provided insight and opportunities for further investigation.

How much of an overhang can we produce by stacking identical rectangular blocks at the edge of a table? It has been known for at least 100 years that the overhang can be as large as desired: we arrange the blocks in the form of a staircase. With $n$ blocks of length 2 the overhang can be made to sum to $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}$. Since the harmonic series diverges, it follows that the overhang can be arranged to be as large as desired, simply by using a suitably large number of blocks. Recently, a number of interesting twists have been added to this paradoxical staircase. I'll be talking about some of these new developments and in particular about a continuous counterpart of the staircase that I've been pondering together with my colleagues David Treeby and Marty Ross.

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.

This talk is a continuation of my talk last August. I will discuss the recently-obtained answers to the open questions I described then.

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.

Tropical geometry is a young field with multiple origins. These include the work of Bergman on logarithmic limit sets of algebraic varieties; the work of the Brazilian computer scientist Simon on discrete mathematics; the work of Bieri, Neumann and Strebel on geometric invariants of groups; and, of course, the work of Newton on polynomials. Even though there is still need for a unified foundation of the field, there is an abundance of applications of tropical geometry in group theory, combinatorics, computational algebra and algebraic geometry. In this talk I will give an overview of (what I understand to be) tropical geometry with a bias towards applications to group theory and low-dimensional topology.

The Weil-Petersson metric is an exciting metric on a space of simple plane curves. In this talk the speaker will introduce the shape space and demonstrate the connection with the Euler-Poincare equations on the group of diffeomorphisms (EPDiff). A numerical method for finding geodesics between two shapes will be demonstrated and applied to the surface of the hippocampus to study the effects of Alzheimer's disease. As another application the speaker will discuss how to do statistics on the shape space and what should be done to improve it.

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.

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

Given a compact Riemann surface X and a point p in X, we construct a holomorphic function without critical points on the punctured (algebraic) Riemann surface R=X-p which is of finite order at the point p. In the case at hand this improves the 1967 theorem of Gunning and Rossi to the effect that every open Riemann surface admits a noncritical holomorphic function, but without any particular growth condition. (Joint work with Takeo Ohsawa.)

We prove that every circle domain in the Riemann sphere admits a proper holomorphic embedding into the affine plane C^2. By a circle domain we mean a domain obtained by removing from the Riemann sphere a finite or countable family of pairwise disjoint closed round discs. Our proof also applies to some circle domains with punctures. The uniformization theorem of He and Schramm (1996) says that every domain in the Riemann sphere with at most countably many boundary components is conformally equivalent to a circle domain, so our theorem embeds all such domains properly holomorphically in C^2. (Joint work with Erlend F. Wold.)

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

Lecture series by Rajesh Gopakumar (Harish-Chandra Research Institute). The lectures will be supplemented by talks by other invited speakers.

Calabi-Yau manifolds are Riemannian manifolds with holonomy group SU(m). They are Ricci-flat and Kahler and admit a 2-parameter family of parallel spinors. In the talk we will discuss the Lorentzian conformal analogue of this situation. If on a manifold a class of conformally equivalent metrics [g] is given, then one can consider the holonomy group of the conformal manifold (M,[g]), which is a subgroup of O(p+1,q+1) if the metric g has signature (p,q). There is a close relation between algebraic properties of the conformal holonomy group and the existence of Einstein metrics in the conformal class as well as to the existence of conformal Killing spinors. In the talk I will explain classification results for conformal holonomy groups of Lorentzian manifolds. In particular, I will describe Lorentzian manifolds (M,g) with conformal holonomy group SU(1,m), which can be viewed as the conformal analogue of Calabi-Yau manifolds. Such Lorentzian metrics g, known as Fefferman metrics, appear on S^1-bundles over strictly pseudoconvex CR spin manifolds and admit a 2-parameter family of conformal Killing spinors.

Lecture series by Jonathan Rosenberg (University of Maryland). There will be additional talks by other invited speakers.

The de Rham complex is of fundamental importance in differential geometry. After first introducing differential forms (in the familiar setting of Euclidean space), I will demonstrate how the de Rham complex elegantly encodes one half (in a sense which will become apparent) of the results from vector calculus. If there is time, I will indicate how results from the remaining half of the theory can be concisely expressed by a single, far more general theorem.

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.

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

In this talk, I will describe a mechanism for determining instability of standing wave solutions to a class of inhomogeneous nonlinear Schrodinger (NLS) equations. The inhomogeneity in this case means that the equations will spatially alternate between NLS and the so-called Gross-Pitaevskii equation. Such equations are useful in 1-D models of Bose-Einstein Condensates (BECs). The mechanism is inherently topological and therefore robust, leading to its application to a number of different soliton solutions, such as gap solitons, surface gap solitons, and dark soliton among others.

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.

We look at an equation arising from the differential-geometric problem of specifying the scalar curvature of a manifold.

Motivated by certain tea room discussions, I am going to (attempt to) model the flow of a viscous fluid under gravity over conical topography. The method used is 'Smoothed Particle Hydrodynamics' (SPH), which is an easy-to-use but perhaps limited-accuracy computational method. The model could be extended to include solidification and thermodynamic effects that can also be implemented within the framework of SPH, and this has the obvious practical application to the modelling of the coverage of ice cream with ice magic, I mean, lava flows. If I fail to achieve this within the next 4 weeks, I will have to go through a talk on SPH that I gave during honours instead.

I will attempt to explain what a totally disconnected, locally compact group is, and then describe some new work with George Willis on an attempt to create new examples based on Baumslag-Solitar groups, which are well known, tried and tested examples/counterexamples in geometric/combinatorial group theory. I will describe how to compute invariants of scale and flat rank for these groups.

Let pk=(N,e) be an RSA public key with corresponding secret key sk=(d,p,q,...). Assume that we obtain partial error-free information of sk, e.g., assume that we obtain half of the most significant bits of p. Then there are well-known algorithms to recover the full secret key. As opposed to these algorithms that allow for correcting erasures of the key sk, we present for the first time a heuristic probabilistic algorithm that is capable of correcting errors in sk provided that e is small. That is, on input of a full but error-prone secret key sk' we reconstruct the original sk by correcting the faults. More precisely, consider an error rate of d in [0,1), where we flip each bit in sk with probability d resulting in an erroneous key sk'. Our Las-Vegas type algorithm allows to recover sk from sk' in expected time polynomial in logN with success probability close to 1, provided that d is strictly less than 0.237. We also obtain a polynomial time Las-Vegas factorization algorithm for recovering the factorization (p,q) from an erroneous version with error rate d strictly less than 0.084.

If you only go to one undergraduate seminar this year, then you should have gone to Jim Denier's - it was cracking, but if you decide to go to another, then this one has cholera, Bayesian statistics, random networks and zombies. Warning: may contain an overuse of pop culture references to motivate an interest in statistics.

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).

Spatial-point data sets, generated from a wide range of physical systems and mathematical models, can be analyzed by counting the number of objects in equally sized bins. We find that the bin counts are related to the Polya distribution. New indexes are developed which quantify whether or not a spatial data set is at its most evenly distributed state. Using three case studies (Lagrangian fluid particles in chaotic laminar flows, cellular automata agents in discrete models, and biological cells within colonies), we calculate the indexes and predict the spatial-state of the system.

In this talk, methane (CH4) adsorption is investigated on both graphite and in the region between two aligned single-walled carbon nanotubes, which we refer to as the groove site. The Lennard–Jones potential function and the continuous approximation is exploited to determine surface binding energies between a single CH4 molecule and graphite and between a single CH4 and two aligned single-walled carbon nanotubes. The modelling indicates that for a CH4 molecule interacting with graphite, the binding energy of the system is minimized when the CH4 carbon is 3.83 angstroms above the surface of the graphitic carbon, while the binding energy of the CH4–groove site system is minimized when the CH4 carbon is 5.17 angstroms away from the common axis shared by the two aligned single-walled carbon nanotubes. These results confirm the current view that for larger groove sites, CH4 molecules in grooves are likely to move towards the outer surfaces of one of the single-walled carbon nanotubes. The results presented in this talk are computationally efficient and are in good agreement with experiments and molecular dynamics simulations, and show that CH4 adsorption on graphite and groove surfaces is more favourable at lower temperatures and higher pressures.

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

In this talk, I will present what I did in my honours project, which was to determine if immigrants, categorised as immigrants from English speaking countries and Non-English speaking countries, are discriminated in the Australian labour market. To determine if discrimination exists, a decomposition of the wage function is applied and analysed via regression analysis. Two different methods of estimating the unknown parameters in the wage function will be discussed: 1. the Ordinary Least Square method, 2. the Quantile Regression method. This is your rare chance of hearing me talk about non-nanomathematics related stuff!

Atiyah-Patodi-Singer proved an index theorem for non-local boundary conditions in the 1970's that has been widely used in mathematics and mathematical physics. A key application of their theory gives the index theorem for signature operators on oriented manifolds with boundary. As a consequence, they defined certain secondary invariants that were metric independent. I will discuss some recent work with Benameur where we extend the APS theory to signature operators twisted by an odd degree closed differential form, and study the corresponding secondary invariants.

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

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.

The idea of continuous symmetry is often described in mathematics via Lie groups. These groups can be classified by their root systems: collections of vectors satisfying certain symmetry properties. The root systems are described in a concise way by Dynkin diagrams, and it turns out, roughly speaking, that there are only seven possible shapes for a Dynkin diagram. In this talk I'll describe some simple examples of Lie groups, explain what a root system is, and show how a Dynkin diagram encodes this information. Then I'll give a very brief sketch of the methods used to classify Dynkin diagrams.

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.

The holonomy group of a semi-Riemannian manifold is defined as the group of parallel transports along loops based at a point. Its connected component, the `restricted holonomy group', is given by restricting in this definition to contractible loops. The restricted holonomy can essentially be described by its Lie algebra and many classification results are obtained in this way. In contrast, the `full' holonomy group is a more global object and classification results are out of reach. In the talk I will describe recent results with H. Baum and K. Laerz (both HU Berlin) about the full holonomy group of so-called `indecomposable' Lorentzian manifolds. I will explain a construction method that arises from analysing the effects on holonomy when dividing the manifold by the action of a properly discontinuous group of isometries and present several examples of Lorentzian manifolds with disconnected holonomy groups.

Recently, the Australian Government has emphasised the need for monitoring and comparing the performance of Australian hospitals. Evaluating the performance of intensive care units (ICUs) is of particular importance, given that the most severe cases are treated in these units. Indeed, ICU performance can be thought of as a proxy for the overall performance of a hospital. We compare the performance of the ICUs contributing to the Australian and New Zealand Intensive Care Society (ANZICS) Adult Patient Database, the largest of its kind in the world, and identify those ICUs with unusual performance. It is well-known that there are many statistical issues that must be accounted for in the evaluation of healthcare provider performance. Indicators of performance must be appropriately selected and estimated, investigators must adequately adjust for casemix, statistical variation must be fully accounted for, and adjustment for multiple comparisons must be made. Our basis for dealing with these issues is the estimation of a hierarchical logistic model for the in-hospital death of each patient, with patients clustered within ICUs. Both patient- and ICU-level covariates are adjusted for, with a random intercept and random coefficient for the APACHE III severity score. Given that we expect most ICUs to have similar performance after adjustment for these covariates, we follow Ohlssen et al., JRSS A (2007), and estimate a null model that we expect the majority of ICUs to follow. This methodology allows us to rigorously account for the aforementioned statistical issues, and accurately identify those ICUs contributing to the ANZICS database that have comparatively unusual performance. This is joint work with Prof. Patty Solomon and Assoc. Prof. John Moran.

In financial derivatives pricing theory, it is very common to change the probability measure from historical measure "real world" into a Risk-Neutral measure as a development of the non arbitrage condition. Girsanov theorem is the most known example of this technique and is used when prices randomness is modelled by Brownian motions. Other genuine candidates for modelling market randomness that have proved efficiency in recent literature are jump process, so how can a change of measure be performed for such processes? This talk will address this question by introducing the non arbitrage condition, discussing Girsanov theorem for diffusion and jump processes and presenting a concrete example.

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

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

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.

Antiviral treatment offers a fast acting alternative to vaccination. It is viewed as a first-line of defence against pandemic influenza, protecting families and household members once infection has been detected. In clinical trials antiviral treatment has been shown to be efficacious in preventing infection, limiting disease and reducing transmission, yet their impact at containing the 2009 influenza A(H1N1)pdm outbreak was limited. I will describe some of our work, which attempts to understand this seeming discrepancy, through the development of a general model and computationally efficient methodology for studying household-based interventions. This is joint work with Dr Andrew Black (Adelaide), and Prof. Matt Keeling and Dr Thomas House (Warwick, U.K.).

A series of four 2-hour lectures by Dr. Ittay Weiss. The theory of dendroidal sets was introduced by Moerdijk and Weiss in 2007 in the study of homotopy operads in algebraic topology. In the five years that have past since then several fundamental and highly non-trivial results were established. For instance, it was established that dendroidal sets provide models for homotopy operads in a way that extends the Joyal-Lurie approach to homotopy categories. It can be shown that dendroidal sets provide new models in the study of n-fold loop spaces. And it is very recently shown that dendroidal sets model all connective spectra in a way that extends the modeling of certain spectra by Picard groupoids. The aim of the lecture series will be to introduce the concepts mentioned above, present the elementary theory, and understand the scope of the results mentioned as well as discuss the potential for further applications. Sources for the course will include the article "From Operads to Dendroidal Sets" (in the AMS volume on mathematical foundations of quantum field theory (also on the arXiv)) and the lecture notes by Ieke Moerdijk "simplicial methods for operads and algebraic geometry" which resulted from an advanced course given in Barcelona 3 years ago. No prior knowledge of operads will be assumed nor any knowledge of homotopy theory that is more advanced then what is required for the definition of the fundamental group. The basics of the language of presheaf categories will be recalled quickly and used freely.

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

We compare spectral and wavelet estimators of the response amplitude operator (RAO) of a linear system, with various input signals and added noise scenarios. The comparison is based on a model of a heaving buoy wave energy device (HBWED), which oscillates vertically as a single mode of vibration linear system. HBWEDs and other single degree of freedom wave energy devices such as the oscillating wave surge convertors (OWSC) are currently deployed in the ocean, making single degree of freedom wave energy devices important systems to both model and analyse in some detail. However, the results of the comparison relate to any linear system. It was found that the wavelet estimator of the RAO offers no advantage over the spectral estimators if both input and response time series data are noise free and long time series are available. If there is noise on only the response time series, only the wavelet estimator or the spectral estimator that uses the cross-spectrum of the input and response signals in the numerator should be used. For the case of noise on only the input time series, only the spectral estimator that uses the cross-spectrum in the denominator gives a sensible estimate of the RAO. If both the input and response signals are corrupted with noise, a modification to both the input and response spectrum estimates can provide a good estimator of the RAO. However, a combination of wavelet and spectral methods is introduced as an alternative RAO estimator. The conclusions apply for autoregressive emulators of sea surface elevation, impulse, and pseudorandom binary sequences (PRBS) inputs. However, a wavelet estimator is needed in the special case of a chirp input where the signal has a continuously varying frequency.

This session will feature the The School of Mathematical Sciences Three Minute Thesis competition. Each postgraduate participating will have three minutes to explain their thesis at a level appropriate for a non-specialist audience. The competition is open to all postgraduates within the School. All staff are welcome to attend.

There are several ways of viewing elements of K^1(X). One of these is via families of unbounded self-adjoint Fredholm operators on X. Each operator will have discrete spectrum, with infinitely many positive and negative eigenvalues of finite multiplicity. One can associate to such a family a geometric object, its graph, and the Chern character and other invariants of the family can be studied from this perspective. By restricting the dimension of the eigenspaces one may sometimes use algebraic topology to completely determine the family up to equivalence. This talk will describe the general framework and some applications to families on low-dimensional manifolds where the methods work well. Various notions related to spectral flow, the index gerbe and Berry phase play roles which will be discussed. This is joint work with Ron Douglas.

In the study of bounded operators on Hilbert spaces of holomorphic functions, concepts and techniques from complex geometry are important. An anti-holomorphic bundle exists on which one can define the Chern connection. Its curvature turns out to be a complete invariant and various operator notions can't be reframed in terms of geometrical ones which leads to the solution of some problems. We will discuss this approach with an emphasis on natural examples in the one and multivariable case.

The speaker will report on a project to develop and promote approaches to mathematics instruction closely related to the Moore method -- methods which are called inquiry-based learning -- as well as on his personal experience of the Moore method. For background, see the speaker's article in the May 2012 issue of the Notices of the American Mathematical Society. To download the article, click on "Media" above.

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 Banach-Tarski Paradox is one of the most counter intuitive results in set theory. It states that a ball can be cut up into a finite number of pieces, which using just rotations and translations can be reassembled into two identical copies of the original ball. This contradicts our naive belief that cutting, rotating and translating objects in Euclidean space should preserve volume. However the construction of the "cutting" is heavily dependent on the axiom of choice, and the resultant pieces are non-measurable, i.e. no consistent notion of volume can be assigned to them. A stronger form of the theorem states that any two bounded subsets of R^3 with non-empty interior are equidecomposable, that is one can be disassembled and reassembled into the other. I'll be going through a brief proof of the theorem (and in doing so further alienate the pure mathematicians in the room from everybody else).

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

The result of an AFL game is always unpredictable - we all know that. Hence why we discuss the weekend's upsets and the local tipping competition as part of the "water-cooler and weekend" conversation on a Monday morning. Different people use various weird and wonderful techniques or criteria to predict the winning team. With readily available data, I will investigate and compare various strategies and define a measure of the hardness of a round (full acknowledgements will be made in my presentation). Hopefully this will help me for next year's tipping competition...

Euclid's geometry describes the world around us in terms of points, lines and planes. For two thousand years these have formed the limited repertoire of basic geometric objects with which to describe the universe. Fractals immeasurably enhance this world-view by providing a description of much around us that is rough and fragmented---of objects that have structure on many sizes.

Let $X$ be a complex algebraic variety defined over the ring $\mathfrak{O}$ of integers in a number field $K$ and let $\Gamma$ be a group of $\mathfrak{O}$-automorphisms of $X$. I shall discuss how the counting of rational points over reductions mod $p$ of $X$, and an analysis of the Hodge structure of the cohomology of $X$, may be used to determine the cohomology as a $\Gamma$-module. This will include some joint work with Alex Dimca and with Mark Kisin, and some classical unsolved problems.

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

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

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.

A group marched on a checkered path,
Bold but split in parts.
They turned and all were lost,
Save five regal hearts.

In this talk, we shall use noncommutative geometry to obtain an index theorem in conformal geometry. This index theorem follows from an explicit and geometric computation of the Connes-Chern character of the spectral triple in conformal geometry, which was introduced recently by Connes and Moscovici. This (twisted) spectral triple encodes the geometry of the group of conformal diffeomorphisms on a spin manifold. The crux of of this construction is the conformal invariance of the Dirac operator. As a result, the Connes-Chern character is intimately related to the CM cocycle of an equivariant Dirac spectral triple. We compute this equivariant CM cocycle by heat kernel techniques. On the way we obtain a new heat kernel proof of the equivariant index theorem for Dirac operators. (Joint work with Raphael Ponge.)

Ever wondered whether you are more likely to die in the Galactic Empire or Middle Earth? Well this is the postgraduate seminar for you! I'll be attempting to answer this question using survival analysis, the statistical method of choice for investigating time to event data. Spoiler Warning: This talk will contain references to the deaths of characters in the above movie sagas.

The game of cricket is unique among ball sports by the ignorant exploitation of \thetitle in the practice of swing bowling, often referred to as a "mysterious art". I will talk a bit about the Magnus effect exploited in inferior sports, the properties of a cricket ball that allow swing bowling, and the explanation of three modes of swing (conventional, contrast and reverse). Following that there will be some discussion on how I plan to use mathematics to turn this "art" into science.

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

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.

Principal Component Analysis (PCA) has become something of a buzzword recently in a number of disciplines including the gene expression and facial recognition. It is a classical, and fundamentally simple, concept that has been around since the early 1900's, its recent popularity largely due to the need for dimension reduction techniques in analyzing high dimensional data that has become more common in the last decade, and the availability of computing power to implement this. I will explain the concept, prove a result, and give a couple of examples. The talk should be accessible to all disciplines as it (should?) only assume first year linear algebra, the concept of a random variable, and covariance.

This aims to be an example of an exemplary talk on examples of celebrated counterexamples in mathematics. A famous example, for example, is Euler's counterexample to Fermat's conjecture in number theory.

A finite graph is called symmetric if its automorphism group is transitive on the set of arcs (ordered pairs of adjacent vertices) of the graph. This is to say that all arcs have the same status in the graph. I will talk about recent results on the classification of a family of symmetric graphs with complete quotients. The most interesting graphs arising from this classification are defined in terms of Hermitian unitals (which are specific block designs), and they admit unitary groups as groups of automorphisms. I will also talk about applications of our results in constructing large symmetric graphs of given degree and diameter. This talk contains joint work with M. Giulietti, S. Marcugini and F. Pambianco.

The ancient Chinese used it, the Celts had this skill in spades, it was a big skill of seafarers and pirates, and even now we need it if only to be able to wear shoes! This talk will be about Knot Theory. Knot theory has a colourful and interesting past and I will touch on the why, the what and the when of knots in mathematics. I shall also discuss the major problems concerning knots including the different methods of classification of knots, the unresolved questions about knots, and why have they even been studied. It will be a thorough immersion that will leave you knotted!

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.

Traditionally, the geometric quantisation of an action by a compact Lie group on a compact symplectic manifold is defined as the equivariant index of a certain Dirac operator. This index is a well-defined formal difference of finite-dimensional representations, since the Dirac operator is elliptic and the manifold and the group in question are compact. From a mathematical and physical point of view however, it is very desirable to extend geometric quantisation to noncompact groups and manifolds. Defining a suitable index is much harder in the noncompact setting, but several interesting results in this direction have been obtained. I will review the difficulties connected to noncompact geometric quantisation, and some of the solutions that have been proposed so far, mainly in connection to the "quantisation commutes with reduction" principle. (An introduction to this principle will be given in my talk at the Colloquium on the same day.)

Many problems within applied mathematics require the solution of a linear system of equations. For instance, models of arterial umbilical blood flow are obtained through a finite element approximation, resulting in a linear, n x n system. For small systems the solution is (almost) trivial, but what happens when n is large? Say, n ~ 10^6? In this case matrix inversion is expensive (read: completely impractical) and we seek approximate solutions in a reasonable time. In this talk I will discuss the basic theory underlying Krylov subspace methods; a class of non-stationary iterative methods which are currently the methods-of-choice for large, sparse, linear systems. In particular I will focus on the method of Generalised Minimum RESiduals (GMRes), which is of the most popular for nonsymmetric systems. It is hoped that through this presentation I will convince you that a) solving linear systems is not necessarily trivial, and that b) my lack of any tangible results is not (entirely) a result of my own incompetence.

From cryptanalysis to a powerful tool which made identity based cryptography possible, pairings have a range of applications in cryptography. I will present basic background (algebraic geometry) needed to understand pairings, hard problems associated with pairings and protocols which use pairings.

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.

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

The analysis of turbulence in transient flows has applications across a broad range of fields. We use the flow of fluid in a toroidal container as a paradigm for studying the complex dynamics due to this turbulence. To explore the dynamics of our system, we exploit the numerical capabilities of semtex; a quadrilateral spectral element DNS code. Rainbows result.

Recently I gave a short talk about how researchers use mathematics to estimate the time since a species' most recent common ancestor. I also pointed out why this generally doesn't work when a population hasn't had a constant population size. Then I quickly changed the subject. In this talk I aim to reintroduce the Coalescent Model, show how it works in general, and finally how researcher's deal with varying a population size.

Clinical trials are the way in which modern medical systems test whether individual treatments are worthwhile. Sophisticated statistics is used to try and make the conclusions from clinical trials as meaningful as possible. What can a very simple probability model then tell us about the worth of multiple treatments? What might the implications of this be for the whole health system?

This talk is based on research currently being conducted with a physician at a major Adelaide hospital. It requires no health knowledge and was not tested on animals. All you need is an enquiring and open mind.

Optimal designs are a class of experimental designs that are optimal with respect to some statistical criterion. That answers the question, right? But what do I mean by 'optimal', and which 'statistical criterion' should you use? In this talk I will answer all these questions, and provide an overly simple example to demonstrate how optimal design works. I will then give a brief explanation of how I will use this methodology, and what chickens have to do with it.

I will give an example of an application of supermanifold theory to physics, i.e., how to "superize" the moduli space of instantons on a 4-fold and use it to give a description of the BRST transformations, to compute the "supermeasure" of the moduli space, and the Nekrasov partition function.

An important component of pathogen evolution at the population level is evolution within hosts, which can alter the composition of genotypes available for transmission as infection progresses. I will present a deterministic multi-scale model, linking the within-host competition dynamics with the transmission dynamics at a population level. I will take HIV as an example of how this framework can help clarify the conflicting evolutionary pressure an infectious disease might be subject to.

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.

A slightly unusual (for this School at least) and hopefully entertaining look at representations of mathematics and mathematicians in popular culture. Do these representations affect people's perceptions of mathematics and its mysterious practitioners? What examples of positive and negative representations are there? Should we care and should it affect our enjoyment those texts? All these questions and many more will remain hopelessly unanswered as we try to cover examples such as Numb3rs, Mean Girls, A Beautiful Mind, Good Will Hunting, 21, The Simpsons and Futurama. Feel free to come prepared with your own examples of egregious crimes against mathematics or refreshing beacons of hope.

Both age and household structure are recognised as important heterogeneities affecting epidemic spread of infectious pathogens, and many models exist nowadays that include either or both forms of heterogeneity. However, different models may fit aggregate epidemic data equally well and nevertheless lead to different predictions of public health interest. I will here present an overview of stochastic epidemic models with increasing complexity in their social structure, focusing in particular on households models. For these models, I will present recent results about the definition and computation of the basic reproduction number R0 and its relationship with other threshold parameters. Finally, I will use these results to compare models with no, either or both age and household structure, with the aim of quantifying the conditions under which each form of heterogeneity is relevant and therefore providing some criteria that can be used to guide model design for real-time predictions.

For each natural number d, the space of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the degree 3 case, studying a double action of the Mobius group on the space of cubic rational maps. We show that the categorical quotient is C, and that the space of cubic rational maps enjoys the holomorphic flexibility properties of strong dominability and C-connectedness.

The study of flow in open helically-wound channels has application to many natural and industrial flows. We will consider laminar flow down helically-wound channels of rectangular cross section and with small torsion, in which the fluid depth is small. Assuming a steady-state flow that is independent of position along the axis of the channel, the flow solution may be determined in the two-dimensional cross section of the channel. A thin-film approximation yields explicit expressions for the fluid velocity in terms of the free-surface shape. The latter satisfies an interesting non-linear ordinary differential equation that, for a channel of rectangular cross section, has an analytical solution. The predictions of the thin-film model are shown to be in good agreement with much more computationally intensive solutions of the small-helix-torsion Navier-Stokes equations. This work has particular relevance to spiral particle separators used in the minerals processing industry. Early work on modelling of particle-laden thin-film flow in spiral channels will also be discussed.

The South Australian electoral system has a history of bias, malapportionment and perceived unfairness. These days, it is typical of most systems across Australia, except with one major difference - a specific legislated criterion designed to force the system to be 'fair'. In reality, fairness is a hard concept to define, and an even harder concept to enforce. In this talk I will briefly take you through the history of South Australian electoral reform, the current state of affairs and my proposed research. There will be very little in the way of rigorous mathematics. No knowledge of politics is assumed, but an understanding of the process of voting would be useful.

Here we investigate the interaction of deoxyribonucleic acid (DNA) inside single walled carbon nanotubes (SWCNTs). Using classical applied mathematical modeling, we derive explicit analytical expressions for the encapsulation of DNA inside single-walled carbon nanotubes. We adopt the 6-12 Lennard-Jones potential function together with the continuous approach to determine the preferred minimum energy position of the dsDNA molecule inside a single-walled carbon nanotube, so as to predict its location with reference to the cross- section of the carbon nanotube. An analytical expression is obtained in terms of hypergeometric functions, which provides a computationally rapid procedure to determine critical numerical values.

We propose a Spatiotemporal Autoregressive Partially Linear Regression ( STARPLR) model for data observed irregularly over space and regularly in time. The model is capable of catching possible non linearity and nonstationarity in space by coefficients to depend on locations. We suggest two-step procedure to estimate both the coefficients and the unknown function, which is readily implemented and can be computed even for large spatio-temoral data sets. As an illustration, we apply our model to analyze the 51 States' House Price Indexes (HPIs) in USA.

We are interested in the dynamics of the microbial population from a copper sulphide bioleaching heap. The composition of the microbial consortium is closely related to the kinetics of the oxidation processes that lead to copper recovery. Using a non-linear model, which considers the effect of substrate depletion and incorporates spatial dependence, we analyse adjacent strips correlation, patterns of microbial succession, relevance of pertinent physic-chemical parameters and the implications of the absence of barriers between the three lifts of the heap. We also explore how the dynamics of the microbial community relate to the mineral composition of the individual strips of the bioleaching pile.

We review Mathai-Wu's recent extension of Ray-Singer analytic torsion to supercomplexes. We explore some new results relating these two torsions, and how we can apply the adiabatic spectral sequence due to Forman and Farber's analytic deformation theory to compute some spectral invariants of the complexes involved, answering some questions that were posed in Mathai-Wu's paper.

Generalized complex geometry combines complex and symplectic geometry into a single framework, incorporating also holomorphic Poisson and bi-Hermitian structures. The Dolbeault complex naturally extends to the generalized complex setting giving rise to Hodge structures in twisted cohomology. We consider the variations of Hodge structure and period mappings that arise from families of generalized complex manifolds. As an application we prove a local Torelli theorem for generalized Calabi-Yau manifolds.

I will give an introduction to a sequence of ideas in tropical geometry, the tropicalization of linear spaces. In the beginning, a construction due to De Concini and Procesi (wonderful models, 1995) gave a combinatorially explicit description of various iterated blowups of projective spaces along (proper transforms of) linear subspaces. A decade later, Tevelev's notion of tropical compactifications led to, in particular, a new view of the wonderful models and their intersection theory in terms of the theory of toric varieties (via work of Feichtner-Sturmfels, Feichtner-Yuzvinsky, Ardila-Klivans, and others). Recently, these ideas have played a role in Huh and Katz's proof of a long-standing conjecture in combinatorics.

A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples of Cayley groups with his Cayley map back in 1846. Over an algebraically closed field of characteristic 0, Cayley and stably Cayley simple groups were classified by Lemire, Popov and Reichstein in 2006. In recent joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of characteristic 0.

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.

Symplectic and projective structures may be compatibly combined. The resulting structure closely resembles conformal geometry and a manifold endowed with such a structure is called conformally Fedosov. This talk will present the basic theory of conformally Fedosov geometry and, in particular, construct a Cartan connection for them. This is joint work with Jan Slovak.

We consider a configuration space of two solids rolling on each other without slipping or twisting, and identify it with an open subset U of R^5, equipped with a generic distribution D of 2-planes. We will discuss symmetry properties of the pair (U,D) and will mention that, in the case of the two solids being balls, when changing the ratio of their radii, the dimension of the group of local symmetries unexpectedly jumps from 6 to 14. This occurs for only one such ratio, and in such case the local group of symmetries of the pair (U,D) is maximal. It is maximal not only among the balls with various radii, but more generally among all (U,D)s corresponding to configuration spaces of two solids rolling on each other without slipping or twisting. This maximal group is isomorphic to the split real form of the exceptional Lie group G2. In the remaining part of the talk we argue how to identify the space U from the pair (U,D) defined above with the bundle T of totally null real 2-planes over a 4-manifold equipped with a split signature metric. We call T the twistor bundle for rolling bodies. We show that the rolling distribution D, can be naturally identified with an appropriately defined twistor distribution on T. We use this formulation of the rolling system to find more surfaces which, when rigidly rolling on each other without slipping or twisting, have the local group of symmetries isomorphic to the exceptional group G2.

I recently found the need to learn a little about what I had naively believed to be an abstruse branch of number theory, but which turns out to be a ubiquitous and intriguing theory. I'll introduce some of the geometry underlying the elementary theory of modular functions and modular forms. We'll look at some pictures and play with sage, time permitting.

We derive extended Lorentz transformations between inertial frames for relative velocities greater than the speed of light, and which are complementary to the Lorentz transformation giving rise to the Einstein special theory of relativity. The new transformations arise from the same mathematical framework as the Lorentz transformation, displaying singular behaviour when the relative velocity approaches the speed of light and generating the same addition law for velocities, but most importantly, do not involve the need to introduce imaginary masses or complicated physics to provide well-defined expressions.

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.

In our effort to determine the most effective way to use geothermal energy this is a left field, yet enticing, idea. Would this produce more 'extra' power than a geothermal plant on its own? Would there be sufficient benefit to interest traditional power generators? We investigated retro-fitting two different geothermal preheating options to a 500MW supercritical steam power plant. We then compared the 'extra-power' produced using geothermal preheating, to the power produced by using geothermal energy on its own. We think the results are interesting and promising, but come along and judge for yourself.

Groups of gauge transformations in quantum field theory are typically extended by a 2-cocycle with values in a certain abelian group due to chiral symmetry breaking. For these extensions there exist a global explicit construction since the 1980's. I shall study the higher group cocycles following a recent paper by F. Wagemann and C. Wockel, but extending to the transformation groupoid setting (motivated by QFT) and discussing potential obstructions in the construction due to a nonvanishing of low dimensional homology groups of the gauge group. The resolution of the obstruction is obtained by an application of the Cheeger-Simons differential characters.

The Kronecker-Weber Theorem has a rich and inspiring history. Much like Fermat's Last Theorem, it can be expressed in a very simple way. Its many proofs often utilise heavy machinery and those who claim it can be solved using elementary means, have quite frankly redefined the meaning of elementary. It has inspired David Hilbert and many other mathematicians leading to a great amount of fantastic work in the area. In this talk, I will discuss this theorem, a 'fairly' simple proof of it as well as discuss how it is relevant to my work and the works of others.

One very important application of number theory is the implementation of public key cryptosystems that we use today. I will introduce elementary number theory, Fermat's theorem and use these to explain how ElGamal encryption and digital signatures work.

I will review my recent work on integrability of M-brane configurations and the description of M-brane models in higher gauge theory. In particular, I will discuss categorified analogues of instantons and present superconformal equations of motion for the non-abelian tensor multiplet in six dimensions. The latter are derived from considering non-abelian gerbes on certain twistor spaces.

Good question. Many functions encountered in practice can be well-approximated by a linear combination of Chebyshev polynomials, which then allows the use of some powerful numerical techniques. I will give a very brief overview of the theory behind some of these methods, demonstrate how they may be implemented using the MATLAB package known as Chebfun, and answer the question posed in the title along the way. No knowledge of approximation theory or MATLAB is required, however, you will need to accept the transliteration "Chebyshev".

Conformal Killing spinors are the solutions of the conformally covariant twistor equation on spinors. Special cases are parallel and Killing spinors, the latter appear as eigenspinors of the Dirac operator on compact Riemannian manifolds of positive scalar curvature for the smallest possible positive eigenvalue. In the talk I will discuss geometric properties of manifolds admitting (conformal) Killing spinors. In particular, I will explain a local classification of the special geometric structures admitting conformal Killing spinors without zeros in the Riemannian as well as in the Lorentzian setting.

Many mathematical and engineering problems have a multiscale nature. There are a vast of theories supporting multiscale modelling on infinite domain, such as homogenization theory and centre manifold theory. To date, there are little consideration of the correct boundary conditions to be used at the edge of macroscale model. In this seminar, I will present how to derive macroscale boundary conditions for the diffusion system.

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.

In mathematical finance, as in many other fields where applied mathematics is a powerful tool, we assume that a model is good enough when it captures different sources of randomness affecting the quantity of interests, which in this case is the electricity prices. The power market is very different from other markets in terms of the randomness sources that can be observed in the prices feature and evolution. We start from suggesting a new model that simulates the electricity prices, this new model is constructed by adding a periodicity term, a jumps terms and a positives mean reverting term. The later term is driven by a non-observable Markov process. So in order to prices some financial product, we have to use some of the filtering theory to deal with the non-observable process, these techniques are gaining very much of interest from practitioners and researchers in the field of financial mathematics.

The category of finite-dimensional smooth manifolds gives rise to interesting structures outside of itself, two examples being mapping spaces and classifying spaces. Diffeological spaces are a notion of generalised smooth space which form a cartesian closed category, so all fibre products and all mapping spaces of smooth manifolds exist as diffeological spaces. Differentiable stacks are a further generalisation that can also deal with moduli spaces (including classifying spaces) for objects with automorphisms. This talk will give an introduction to this circle of ideas.

Colour is a powerful tool in presenting data, but it can be tricky to choose just the right colours to represent your data honestly - do the colours used in your heatmap overemphasise the differences between particular values over others? does your choice of colours overemphasize one when they should be represented as equal? etc. All these questions are fundamentally based in how we perceive colour. There has been alot of research into how we perceive colour in the past century, and some interesting results. I will explain how a `standard observer' was found empirically and used to develop an absolute reference standard for colour in 1931. How although the common Red-Green-Blue representation of colour is useful and intuitive, distances between colours in this space do not reflect our perception of difference between colours and how alternative, perceptually focused colourspaces where introduced in 1976. I will go on to explain how these results can be used to provide simple mechanisms by which to choose colours that satisfy particular properties such as being equally different from each other, or being linearly more different in sequence, or maintaining such properties when transferred to greyscale, or for a colourblind person.

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.

Blood flow within the human arterial system is inherently unsteady as a consequence of the pulsations of the heart. The unsteady nature of the flow gives rise to a number of important flow features which may be critical in understanding pathologies of the cardiovascular system. For example, it is believed that large oscillations in wall shear stress may enhance the effects of artherosclerosis, among other pathologies. In this talk I will present some of the basic concepts of pulsatile flow and follow the analysis first performed by J.R. Womersley in his seminal 1955 paper.

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!

The theory of crystallographic groups acting cocompactly on Euclidean space can be extended and generalised in many different ways. For example, instead of studying discrete groups of Euclidean isometries, one can consider groups of isometries for indefinite inner products. These are the fundamental groups of compact flat pseudo-Riemannian manifolds. Still more generally, one might study group of affine transformation on n-space that are not required to preserve any bilinear form. Also, the condition of cocompactness can be dropped. In this talk, I will present some of the results obtained for these generalisations, and also discuss some of my own work on flat homogeneous pseudo-Riemannian spaces.

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

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.

On doubling metric measure spaces endowed with a Dirichlet form and satisfying the Davies-Gaffney estimate, we show some characterisations of pointwise upper bounds of the heat kernel in terms of one-parameter weighted inequalities which correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new and simpler proof of the well-known equivalence between classical heat kernel upper bounds and the relative Faber-Krahn inequalities. We are also able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth. This is a joint work with Salahaddine Boutayeb and Adam Sikora.

This interdisciplinary workshop will be about aspects of representation theory (in the sense of Harish-Chandra), aspects of noncommutative geometry (in the sense of Alain Connes) and aspects of operator K-theory (in the sense of Gennadi Kasparov). It features the renowned speaker, Professor Nigel Higson (Penn State University) http://www.iga.adelaide.edu.au/workshops/WorkshopJuly2013/ All are welcome.

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.