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Search the School of Mathematical SciencesPeople matching "The index theorem for projective families of ellip" 
Professor Mathai Varghese Elder Professor of Mathematics, Australian Laureate Fellow, Fellow of the Australian Academy of Scie
More about Mathai Varghese... 
Events matching "The index theorem for projective families of ellip" 
A Bivariate Zeroinflated Poisson Regression Model and application to some Dental Epidemiological data 14:10 Fri 27 Oct, 2006 :: G08 Mathematics Building University of Adelaide :: University Prof Sudhir Paul
Data in the form of paired (pretreatment, posttreatment) counts arise in the study of the effects of several treatments after accounting for possible covariate effects. An example of such a data set comes from a dental epidemiological study in Belo Horizonte (the Belo Horizonte caries prevention study) which evaluated various programmes for reducing caries. Also, these data may show extra pairs of zeros than can be accounted for by a simpler model, such as, a bivariate Poisson regression model. In such situations we propose to use a zeroinflated bivariate Poisson regression (ZIBPR) model for the paired (pretreatment, posttreatment) count data. We develop EM algorithm to obtain maximum likelihood estimates of the parameters of the ZIBPR model. Further, we obtain exact Fisher information matrix of the maximum likelihood estimates of the parameters of the ZIBPR model and develop a procedure for testing treatment effects. The procedure to detect treatment effects based on the ZIBPR model is compared, in terms of size, by simulations, with an earlier procedure using a zeroinflated Poisson regression (ZIPR) model of the posttreatment count with the pretreatment count treated as a covariate. The procedure based on the ZIBPR model holds level most effectively. A further simulation study indicates good power property of the procedure based on the ZIBPR model. We then compare our analysis, of the decayed, missing and filled teeth (DMFT) index data from the caries prevention study, based on the ZIBPR model with the analysis using a zeroinflated Poisson regression model in which the pretreatment DMFT index is taken to be a covariate 

Finite Geometries: Classical Problems and Recent Developments 15:10 Fri 20 Jul, 2007 :: G04 Napier Building University of Adelaide :: Prof. Joseph A. Thas :: Ghent University, Belgium
In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory, cryptography and design of experiments have made the field even more attractive. In my talk some classical problems and recent developments will be discussed. First I will mention Segre's celebrated theorem and ovals and a purely combinatorial characterization of Hermitian curves in the projective plane over a finite field here, from the beginning, the considered pointset is contained in the projective plane over a finite field. Next, a recent elegant result on semiovals in PG(2,q), due to GÃ¡cs, will be given. A second approach is where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space. This will be illustrated by a characterization of the classical inversive plane in the odd case. Another quite recent beautiful result in Galois geometry is the discovery of an infinite class of hemisystems of the Hermitian variety in PG(3,q^2), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for GF(9), due to Segre, was known. 

An Introduction to invariant differential pairings 14:10 Tue 24 Jul, 2007 :: Mathematics G08 :: Jens Kroeske
On homogeneous spaces G/P, where G is a semisimple Lie group and P is a
parabolic subgroup (the ordinary sphere or projective spaces being
examples), invariant operators, that is operators between certain
homogeneous bundles (functions, vector fields or forms being amongst the
typical examples) that are invariant under the action of the group G, have
been studied extensively. Especially on so called hermitian symmetric spaces
which arise through a 1grading of the Lie algebra of G there exists a
complete classification of first order invariant linear differential
operators even on more general manifolds (that allow a so called almost
hermitian structure).
This talk will introduce the notion of an invariant bilinear differential
pairing between sections of the aforementioned homogeneous bundles. Moreover
we will discuss a classification (excluding certain totally degenerate
cases) of all first order invariant bilinear differential pairings on
manifolds with an almost hermitian symmetric structure. The similarities and
connections with the linear operator classification will be highlighted and
discussed.


Div, grad, curl, and all that 15:10 Fri 10 Aug, 2007 :: G08 Mathematics Building University of Adelaide :: Prof. Mike Eastwood :: School of Mathematical Sciences, University of Adelaide
These wellknown differential operators are, of course, important in applied mathematics. This is just the tip of an iceberg. I shall indicate some of what lies beneath the surface. There are links with topology, physics, symmetry groups, finite element schemes, and more besides. This talk will touch on these different topics by means of examples. Little prior knowledge will be assumed beyond the equality of mixed partial derivatives. 

Fermat's Last Theorem and modular elliptic curves 15:10 Wed 5 Sep, 2007 :: G08 Mathematics Building University of Adelaide :: Dr Mark Kisin
Media...I will give a historical talk, explaining the steps by which one can deduce Fermat's Last Theorem from a statement about modular forms and elliptic curves. 

Betti's Reciprocal Theorem for Inclusion and Contact Problems 15:10 Fri 1 Aug, 2008 :: G03 Napier Building University of Adelaide :: Prof. Patrick Selvadurai :: Department of Civil Engineering and Applied Mechanics, McGill University
Enrico Betti (18231892) is recognized in the mathematics community for his pioneering contributions to topology. An equally important contribution is his formulation of the reciprocity theorem applicable to elastic bodies that satisfy the classical equations of linear elasticity. Although James Clerk Maxwell (18311879) proposed a law of reciprocal displacements and rotations in 1864, the contribution of Betti is acknowledged for its underlying formal mathematical basis and generality. The purpose of this lecture is to illustrate how Betti's reciprocal theorem can be used to full advantage to develop compact analytical results for certain contact and inclusion problems in the classical theory of elasticity. Inclusion problems are encountered in number of areas in applied mechanics ranging from composite materials to geomechanics. In composite materials, the inclusion represents an inhomogeneity that is introduced to increase either the strength or the deformability characteristics of resulting material. In geomechanics, the inclusion represents a constructed material region, such as a ground anchor, that is introduced to provide load transfer from structural systems. Similarly, contact problems have applications to the modelling of the behaviour of indentors used in materials testing to the study of foundations used to distribute loads transmitted from structures. In the study of conventional problems the inclusions and the contact regions are directly loaded and this makes their analysis quite straightforward. When the interaction is induced by loads that are placed exterior to the indentor or inclusion, the direct analysis of the problem becomes inordinately complicated both in terns of formulation of the integral equations and their numerical solution. It is shown by a set of selected examples that the application of Betti's reciprocal theorem leads to the development of exact closed form solutions to what would otherwise be approximate solutions achievable only through the numerical solution of a set of coupled integral equations. 

Elliptic equation for diffusionadvection flows 15:10 Fri 15 Aug, 2008 :: G03 Napier Building University of Adelaide :: Prof. Pavel Bedrikovsetsky :: Australian School of Petroleum Science, University of Adelaide.
The standard diffusion equation is obtained by Einstein's method and its generalisation, FokkerPlankKolmogorovFeller theory. The time between jumps in Einstein derivation is constant.
We discuss random walks with residence time distribution, which occurs for flows of solutes and suspensions/colloids in porous media, CO2 sequestration in coal mines, several processes in chemical, petroleum and environmental engineering. The rigorous application of the Einstein's method results in new equation, containing the time and the mixed dispersion terms expressing the dispersion of the particle time steps.
Usually, adding the second time derivative results in additional initial data. For the equation derived, the condition of limited solution when time tends to infinity provides with uniqueness of the Caushy problem solution.
The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advectiondispersion) equation. This is in agreement with the experimental observations and predictions of the direct simulation.


On the HenstockKurzweil integral (along with concerns about general math education in Europe) 15:10 Fri 13 Feb, 2009 :: Napier LG28 :: Prof JeanPierre Demailly :: University of Grenoble, France
The talk will be the occasion to take a few minutes to describe the situation of math education in France and in Europe, to motivate the interest of the lecturer in trying to bring back rigorous proofs in integration theory. The remaining 45 minutes will be devoted to explaining the basics of HenstockKurzweil integration theory, which, although not a response to education problems, is a modern and elementary approach of a very strong extension of the Riemann integral, providing easy access to several fundamental results of Lebesgue theory (monotone convergence theorem, existence of Lebesgue measure, etc.). 

The index theorem for projective families of elliptic operators 13:10 Fri 13 Mar, 2009 :: School Board Room :: Prof Mathai Varghese :: University of Adelaide


Boltzmann's Equations for Suspension Flow in Porous Media and Correction of the Classical Model 15:10 Fri 13 Mar, 2009 :: Napier LG29 :: Prof Pavel Bedrikovetsky :: University of Adelaide
Suspension/colloid transport in porous media is a basic phenomenon in environmental, petroleum and chemical engineering. Suspension of particles moves through porous media and particles are captured by straining or attraction. We revise the classical equations for particle mass balance and particle capture kinetics and show its nonrealistic behaviour in cases of large dispersion and of flowfree filtration. In order to resolve the paradoxes, the porescale model is derived. The model can be transformed to Boltzmann equation with particle distribution over pores. Introduction of sinksource terms into Boltzmann equation results in much more simple calculations if compared with the traditional ChapmanEnskog averaging procedure. Technique of projecting operators in Hilbert space of Fourier images is used. The projection subspace is constructed in a way to avoid dependency of averaged equations on sinksource terms. The averaging results in explicit expressions for particle flux and capture rate. The particle flux expression describes the effect of advective particle velocity decrease if compared with the carrier water velocity due to preferential capture of "slow" particles in small pores. The capture rate kinetics describes capture from either advective or diffusive fluxes. The equations derived exhibit positive advection velocity for any dispersion and particle capture in immobile fluid that resolves the abovementioned paradox.
Finally, we discuss validation of the model for propagation of contaminants in aquifers, for filtration, for potable water production by artesian wells, for formation damage in oilfields. 

Geometric analysis on the noncommutative torus 13:10 Fri 20 Mar, 2009 :: School Board Room :: Prof Jonathan Rosenberg :: University of Maryland
Noncommutative geometry (in the sense of Alain Connes) involves
replacing a conventional space by a "space" in which the algebra of
functions is noncommutative. The simplest truly nontrivial
noncommutative manifold is the noncommutative 2torus, 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). 

Lagrangian fibrations on holomorphic symplectic manifolds I: Holomorphic Lagrangian fibrations 13:10 Fri 5 Jun, 2009 :: School Board Room :: Dr Justin Sawon :: Colorado State University
A compact K{\"a}hler manifold $X$ is a holomorphic symplectic manifold if it admits a nondegenerate holomorphic twoform $\sigma$. 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. 

Generalizations of the SteinTomas restriction theorem 13:10 Fri 7 Aug, 2009 :: School Board Room :: Prof Andrew Hassell :: Australian National University
The SteinTomas 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 LaplaceBeltrami operator on
certain complete Riemannian manifolds. It turns out that dynamical
properties of the geodesic flow play a crucial role in determining whether
a restrictiontype theorem holds for these manifolds.


Defect formulae for integrals of pseudodifferential symbols:
applications to dimensional regularisation and index theory 13:10 Fri 4 Sep, 2009 :: School Board Room :: Prof Sylvie Paycha :: Universite Blaise Pascal, ClermontFerrand, France
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 "cutoff" 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 Monster 12:10 Thu 10 Sep, 2009 :: Napier 210 :: Dr David Parrott :: University of Adelaide
Media...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.


Irreducible subgroups of SO(2,n) 13:10 Fri 16 Oct, 2009 :: School Board Room :: Dr Thomas Leistner :: University of Adelaide
Berger's classification of irreducibly represented Lie groups that can occur as holonomy groups of semiRiemannian 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 KarpelevichMostov theorem. (This is joint work with Antonio J. Di Scala, Politecnico di Torino.) 

Analytic torsion for twisted de Rham complexes 13:10 Fri 30 Oct, 2009 :: School Board Room :: Prof Mathai Varghese :: University of Adelaide
We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by RaySinger, 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 Tdual circle bundles with closed 3form flux. This is joint work with Siye Wu. 

Hartogstype holomorphic extensions 13:10 Tue 15 Dec, 2009 :: School Board Room :: Prof Roman Dwilewicz :: Missouri University of Science and Technology
We will review holomorphic extension problems starting with the famous Hartogs extension theorem (1906), via SeveriKneserFicheraMartinelli 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 CauchyRiemann functions are fundamental problems in complex analysis of several variables. The talk will be very elementary, with many figures, and accessible to graduate and even advanced undergraduate students. 

A solution to the GromovVaserstein problem 15:10 Fri 29 Jan, 2010 :: Engineering North N 158 Chapman Lecture Theatre :: Prof Frank Kutzschebauch :: University of Berne, Switzerland
Any matrix in $SL_n (\mathbb C)$ can be written as a product of elementary matrices using the Gauss elimination process. If instead of the field of complex numbers, the entries in the matrix are elements of a more general ring, this becomes a delicate question. In particular, rings of complexvalued functions on a space are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in $m$ variables in case the size $n$ of the matrix is at least 3. In the topological category, the problem was solved by Thurston and Vaserstein. For holomorphic functions on $\mathbb C^m$, the problem was posed by Gromov in the 1980s. We report on a complete solution to Gromov's problem. A main tool is the OkaGrauertGromov hprinciple in complex analysis. Our main theorem can be formulated as follows: In the absence of obvious topological obstructions, the Gauss elimination process can be performed in a way that depends holomorphically on the matrix. This is joint work with Bj\"orn Ivarsson. 

Proper holomorphic maps from strongly pseudoconvex domains to qconvex manifolds 13:10 Fri 5 Feb, 2010 :: School Board Room :: Prof Franc Forstneric :: University of Ljubljana
(Joint work with B. Drinovec Drnovsek, Amer. J. Math., in press.)
I will discuss the existence of closed complex subvarieties
of a complex manifold X that are proper holomorphic images
of strongly pseudoconvex Stein domains. The main
sufficient condition is expressed in terms of
the Morse indices and of the number of positive
Levi eigenvalues of an exhaustion function on X.
Examples show that our condition cannot be weakened in general.
I will describe optimal results for subvarieties of this type in
complements of compact complex submanifolds with Griffiths
positive normal bundle; in the projective case these
generalize classical theorems of Remmert, Bishop and
Narasimhan concerning proper holomorphic maps and embeddings
to complex Euclidean spaces. 

Exploratory experimentation and computation 15:10 Fri 16 Apr, 2010 :: Napier LG29 :: Prof Jonathan Borwein :: University of Newcastle
Media...The mathematical research community is facing a great challenge to reevaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to datamine on the Internet. Add to that the enormous complexity of many modern capstone results such as the Poincare conjecture, Fermat's last theorem, and the Classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished. I shall look at the philosophical context with examples and then offer some of five benchmarking examples of the opportunities and challenges we face. 

Understanding convergence of meshless methods: Vortex methods and smoothed particle hydrodynamics 15:10 Fri 14 May, 2010 :: Santos Lecture Theatre :: A/Prof Lou Rossi :: University of Delaware
Meshless methods such as vortex methods (VMs) and smoothed particle
hydrodynamics (SPH) schemes offer many advantages in fluid flow computations.
Particlebased 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 NavierStokes
computations of high Reynolds number flows using BlobFlow, an open
source implementation of the high precision vortex method. 

Topological chaos in two and three dimensions 15:10 Fri 18 Jun, 2010 :: Santos Lecture Theatre :: Dr Matt Finn :: School of Mathematical Sciences
Research into twodimensional 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, traintracks, dynamical systems and
topological index formulae. I will then make some speculations about
topological chaos in three dimensions. 

Index theory in the noncommutative world 13:10 Fri 20 Aug, 2010 :: Ingkarni Wardli B20 (Suite 4) :: Prof Alan Carey :: Australian National University
The aim of the talk is to give an overview of the noncommutative geometry approach to index theory. 

Index theory in Mathematics and Physics 15:10 Fri 20 Aug, 2010 :: Napier G04 :: Prof Alan Carey :: Australian National University
This lecture is a personal (and partly historical) overview in nontechnical terms of the topic described in the title, from first year linear algebra to von Neumann algebras. 

A classical construction for simplicial sets revisited 13:10 Fri 27 Aug, 2010 :: Ingkarni Wardli B20 (Suite 4) :: Dr Danny Stevenson :: University of Glasgow
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 wellknown, some notsowellknown. We will
reexamine the proof of an old theorem of Kan in light of these
functors. We will try to keep all jargon to a minimum. 

Contraction subgroups in locally compact groups 13:10 Fri 17 Sep, 2010 :: Ingkarni Wardli B20 (Suite 4) :: Prof George Willis :: University of Newcastle
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 nontriviality 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). 

IGAAMSI Workshop: Dirac operators in geometry, topology, representation theory, and physics 10:00 Mon 18 Oct, 2010 :: 7.15 Ingkarni Wardli :: Prof Dan Freed :: University of Texas, Austin
Lecture Series by Dan Freed (University of Texas, Austin).
Dirac introduced his eponymous operator to describe electrons in quantum theory.
It was rediscovered by Atiyah and Singer in their study of the index problem on
manifolds. In these lectures we explore new theorems and applications. Several
of these also involve Ktheory 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/ 

Complete quaternionic Kahler manifolds associated to cubic polynomials 13:10 Fri 11 Feb, 2011 :: Ingkarni Wardli B18 :: Prof Vicente Cortes :: University of Hamburg
We prove that the supergravity r and cmaps 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.


Real analytic sets in complex manifolds I: holomorphic closure dimension 13:10 Fri 4 Mar, 2011 :: Mawson 208 :: Dr Rasul Shafikov :: University of Western Ontario
After a quick introduction to real and complex analytic sets,
I will discuss possible notions of complex dimension of real sets, and then discuss a structure theorem for the holomorphic closure dimension which is defined as the dimension of the smallest complex analytic germ containing the real germ. 

A strong Oka principle for embeddings of some planar domains into CxC*, I 13:10 Fri 6 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide
The Oka principle refers to a collection of results in
complex analysis which state that there are only topological
obstructions to solving certain holomorphically defined problems
involving Stein manifolds. For example, a basic version of Gromov's
Oka principle states that every continuous map from a Stein manifold
into an elliptic complex manifold is homotopic to a holomorphic map.
In these two talks I will discuss a new result showing that
if we restrict the class of source manifolds to circular domains and
fix the target as CxC* we can obtain a much stronger Oka principle:
every continuous map from a circular domain S into CxC* is homotopic
to a proper holomorphic embedding. This result has close links with
the longstanding and difficult problem of finding proper holomorphic
embeddings of Riemann surfaces into C^2, with additional motivation
from other sources.


A strong Oka principle for embeddings of some planar domains into CxC*, II 13:10 Fri 13 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide
The Oka principle refers to a collection of results in
complex analysis which state that there are only topological
obstructions to solving certain holomorphically defined problems
involving Stein manifolds. For example, a basic version of Gromov's
Oka principle states that every continuous map from a Stein manifold
into an elliptic complex manifold is homotopic to a holomorphic map.
In these two talks I will discuss a new result showing that
if we restrict the class of source manifolds to circular domains and
fix the target as CxC* we can obtain a much stronger Oka principle:
every continuous map from a circular domain S into CxC* is homotopic
to a proper holomorphic embedding. This result has close links with
the longstanding and difficult problem of finding proper holomorphic
embeddings of Riemann surfaces into C^2, with additional motivation
from other sources.


The ExtendedDomainEigenfunction Method: making old mathematics work for new problems 15:10 Fri 13 May, 2011 :: 7.15 Ingkarni Wardli :: Prof Stan Miklavcic :: University of South Australia
Media...Standard analytical solutions to elliptic boundary value problems on asymmetric domains are rarely, if ever, obtainable. Several years ago I proposed a solution technique to cope with such complicated domains. It involves the embedding of the original domain into one with simple boundaries where the classical eigenfunction solution approach can be used. The solution in the larger domain, when restricted to the original domain is then the solution of the original boundary value problem. In this talk I will present supporting theory for this idea, some numerical results for the particular case of the Laplace equation and the Stokes flow equations in twodimensions and discuss advantages and limitations of the proposal. 

Modelling computer network topologies through optimisation 12:10 Mon 1 Aug, 2011 :: 5.57 Ingkarni Wardli :: Mr Rhys Bowden :: University of Adelaide
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. 

Dirac operators on classifying spaces 13:10 Fri 28 Oct, 2011 :: B.19 Ingkarni Wardli :: Dr Pedram Hekmati :: University of Adelaide
The Dirac operator was introduced by Paul Dirac in 1928 as the formal square
root of the D'Alembert operator. Thirty years later it was rediscovered in
Euclidean signature by Atiyah and Singer in their seminal work on index theory.
In this talk I will describe efforts to construct a Dirac type operator on the
classifying space for odd complex Ktheory. Ultimately the aim is to produce a
projective family of Fredholm operators realising elements in twisted Ktheory
of a certain moduli stack. 

Noncritical holomorphic functions of finite growth on algebraic Riemann surfaces 13:10 Fri 3 Feb, 2012 :: B.20 Ingkarni Wardli :: Prof Franc Forstneric :: University of Ljubljana
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=Xp
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.) 

Embedding circle domains into the affine plane C^2 13:10 Fri 10 Feb, 2012 :: B.20 Ingkarni Wardli :: Prof Franc Forstneric :: University of Ljubljana
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.) 

Forecasting electricity demand distributions using a semiparametric additive model 15:10 Fri 16 Mar, 2012 :: B.21 Ingkarni Wardli :: Prof Rob Hyndman :: Monash University
Media...Electricity demand forecasting plays an important role in shortterm load allocation and longterm planning for future generation facilities and transmission augmentation. Planners must adopt a probabilistic view of potential peak demand levels, therefore density forecasts (providing estimates of the full probability distributions of the possible future values of the demand) are more helpful than point forecasts, and are necessary for utilities to evaluate and hedge the financial risk accrued by demand variability and forecasting uncertainty.
Electricity demand in a given season is subject to a range of uncertainties, including underlying population growth, changing technology, economic conditions, prevailing weather conditions (and the timing of those conditions), as well as the general randomness inherent in individual usage. It is also subject to some known calendar effects due to the time of day, day of week, time of year, and public holidays.
I will describe a comprehensive forecasting solution designed to take all the available information into account, and to provide forecast distributions from a few hours ahead to a few decades ahead. We use semiparametric additive models to estimate the relationships between demand and the covariates, including temperatures, calendar effects and some demographic and economic variables. Then we forecast the demand distributions using a mixture of temperature simulation, assumed future economic scenarios, and residual bootstrapping. The temperature simulation is implemented through a new seasonal bootstrapping method with variable blocks.
The model is being used by the state energy market operators and some electricity supply companies to forecast the probability distribution of electricity demand in various regions of Australia. It also underpinned the Victorian Vision 2030 energy strategy. 

The de Rham Complex 12:10 Mon 19 Mar, 2012 :: 5.57 Ingkarni Wardli :: Mr Michael Albanese :: University of Adelaide
Media...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. 

Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds 13:10 Fri 4 May, 2012 :: Napier LG28 :: Dr Tyson Ritter :: University of Adelaide
In complex geometry a manifold is Stein if there are, in a certain
sense, "many" holomorphic maps from the manifold into C^n. While this
has long been well understood, a fruitful definition of the dual
notion has until recently been elusive. In Oka theory, a manifold is
Oka if it satisfies several equivalent definitions, each stating that
the manifold has "many" holomorphic maps into it from C^n. Related to
this is the geometric condition of ellipticity due to Gromov, who
showed that it implies a complex manifold is Oka.
We present recent contributions to three open questions involving
elliptic and Oka manifolds. We show that affine quotients of C^n are
elliptic, and combine this with an example of Margulis to construct
new elliptic manifolds of interesting homotopy types. It follows that
every open Riemann surface properly acyclically embeds into an
elliptic manifold, extending an existing result for open Riemann
surfaces with abelian fundamental group.


Index type invariants for twisted signature complexes 13:10 Fri 11 May, 2012 :: Napier LG28 :: Prof Mathai Varghese :: University of Adelaide
AtiyahPatodiSinger proved an index theorem for nonlocal 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. 

The change of probability measure for jump processes 12:10 Mon 28 May, 2012 :: 5.57 Ingkarni Wardli :: Mr Ahmed Hamada :: University of Adelaide
Media...In financial derivatives pricing theory, it is very common to change the probability measure from historical measure "real world" into a RiskNeutral 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. 

Enhancing the Jordan canonical form 15:10 Fri 1 Jun, 2012 :: B.21 Ingkarni Wardli :: A/Prof Anthony Henderson :: The University of Sydney
Media...In undergraduate linear algebra, we teach the Jordan canonical form theorem:
that every similarity class of n x n complex matrices contains a special
matrix which is blockdiagonal with each block having a very simple form (a single eigenvalue repeated down the diagonal,
ones on the superdiagonal, and zeroes elsewhere). This is of course very
useful for matrix calculations.
After explaining some of the general context of this result,
I will focus on a case which, despite its close proximity to the Jordan
canonical form theorem, has only recently been worked out: the classification
of pairs of a vector and a matrix.


Epidemiological consequences of householdbased antiviral prophylaxis for pandemic influenza 14:10 Fri 8 Jun, 2012 :: 7.15 Ingkarni Wardli :: Dr Joshua Ross :: The University of Adelaide
Media...Antiviral treatment offers a fast acting alternative to vaccination. It is viewed as a firstline 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 householdbased interventions.
This is joint work with Dr Andrew Black (Adelaide), and Prof. Matt Keeling and Dr Thomas House (Warwick, U.K.). 

Ktheory and unbounded Fredholm operators 13:10 Mon 9 Jul, 2012 :: Ingkarni Wardli B19 :: Dr Jerry Kaminker :: University of California, Davis
There are several ways of viewing elements of K^1(X). One
of these is via families of unbounded selfadjoint 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 lowdimensional 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.


Complex geometry and operator theory 14:10 Mon 9 Jul, 2012 :: Ingkarni Wardli B19 :: Prof Ron Douglas :: Texas A&M University
In the study of bounded operators on Hilbert spaces of holomorphic functions, concepts and techniques from complex geometry are important. An antiholomorphic 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 Four Colour Theorem 11:10 Mon 23 Jul, 2012 :: B.17 Ingkarni Wardli :: Mr Vincent Schlegel :: University of Adelaide
Media...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 graphtheoretic techniques used in the 1976 AppelHaken proof.


The BanachTarski Paradox 11:10 Mon 30 Jul, 2012 :: G.07 Engineering Mathematics :: Mr William Crawford :: University of Adelaide
Media...The BanachTarski 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 nonmeasurable, 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 nonempty 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). 

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

Continuous random walk models for solute transport in porous media 15:10 Fri 17 Aug, 2012 :: B.21 Ingkarni Wardli :: Prof Pavel Bedrikovetski :: The University of Adelaide
Media...The classical diffusion (thermal conductivity) equation was derived from the Master random walk equation and is parabolic. The main assumption was a probabilistic distribution of the jump length while the jump time is constant. Distribution of the jump time along with the jump length adds the second time derivative into the averaged equations, but the equation becomes ... elliptic! Where from to take an extra initial condition? We discuss how to pose the wellposed flow problem, exact 1d solution and numerous engineering applications. This is joint work with A. Shapiro and H. Yuan. 

Noncommutative geometry and conformal geometry 13:10 Fri 24 Aug, 2012 :: Engineering North 218 :: Dr Hang Wang :: Tsinghua University
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 ConnesChern 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 ConnesChern 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.) 

Holomorphic flexibility properties of compact complex surfaces 13:10 Fri 31 Aug, 2012 :: Engineering North 218 :: A/Prof Finnur Larusson :: University of Adelaide
I will describe recent joint work with Franc Forstneric (arXiv, July 2012). We introduce a new property, called the stratified Oka property, which fits into a hierarchy of antihyperbolicity 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 antihyperbolicity 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 EnriquesKodaira classification of surfaces. 

Geometric quantisation in the noncompact setting 13:10 Fri 14 Sep, 2012 :: Engineering North 218 :: Dr Peter Hochs :: Leibniz University, Hannover
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 welldefined formal difference of finitedimensional 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.)


Variation of Hodge structure for generalized complex manifolds 13:10 Fri 7 Dec, 2012 :: Ingkarni Wardli B20 :: Dr David Baraglia :: University of Adelaide
Generalized complex geometry combines complex and symplectic geometry into a single framework, incorporating also holomorphic Poisson and biHermitian 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 CalabiYau manifolds. 

Hyperplane arrangements and tropicalization of linear spaces 10:10 Mon 17 Dec, 2012 :: Ingkarni Wardli B17 :: Dr Graham Denham :: University of Western Ontario
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 FeichtnerSturmfels, FeichtnerYuzvinsky, ArdilaKlivans, and others). Recently, these ideas have played a role in Huh and Katz's proof of a longstanding conjecture in combinatorics. 

Recent results on holomorphic extension of functions on unbounded domains in C^n 11:10 Fri 21 Dec, 2012 :: Ingkarni Wardli B19 :: Prof Roman Dwilewicz :: Missouri University of Science and Technology
In the talk there will be given a short review of holomorphic
extension problems starting with the famous Hartogs theorem (1906) up to recent results on global holomorphic extensions for unbounded domains, obtained together with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago). There is an interesting geometry behind the extension problem for unbounded domains, namely (in some cases) it depends on the position of a complex variety in the closure of the domain. The extension problem appeared nontrivial 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.


Conformally Fedosov manifolds 12:10 Fri 8 Mar, 2013 :: Ingkarni Wardli B19 :: Prof Michael Eastwood :: Australian National University
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. 

On the chromatic number of a random hypergraph 13:10 Fri 22 Mar, 2013 :: Ingkarni Wardli B21 :: Dr Catherine Greenhill :: University of New South Wales
A hypergraph is a set of vertices and a set of hyperedges, where each
hyperedge is a subset of vertices. A hypergraph is runiform 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
kcolouring.
We consider the problem of kcolouring a random runiform hypergraph
with n vertices and cn edges, where k, r and c are constants and n tends
to infinity. In this setting, Achlioptas and Naor showed that for the
case of r = 2, the chromatic number of a random graph must have one of two
easily computable values as n tends to infinity.
I will describe some joint work with Martin Dyer (Leeds) and Alan Frieze
(Carnegie Mellon), in which we generalised this result to random uniform
hypergraphs. The argument uses the second moment method, and applies a
general theorem for performing Laplace summation over a lattice. So the
proof contains something for everyone, with elements from combinatorics,
analysis and algebra. 

A stability theorem for elliptic Harnack inequalities 15:10 Fri 5 Apr, 2013 :: B.18 Ingkarni Wardli :: Prof Richard Bass :: University of Connecticut
Media...Harnack inequalities are an important tool in probability theory,
analysis, and partial differential equations. The classical Harnack
inequality is just the one you learned in your graduate complex analysis
class, but there have been many extensions, to different spaces, such as
manifolds, fractals, infinite graphs, and to various sorts of elliptic operators.
A landmark result was that of Moser in 1961, where he proved the Harnack
inequality for solutions to a class of partial differential equations.
I will talk about the stability of Harnack inequalities. The main result
says that if the Harnack inequality holds for an operator on a space,
then the Harnack inequality will also hold for a large class of other operators
on that same space. This provides a generalization of the result of Moser. 

KroneckerWeber Theorem 12:10 Mon 8 Apr, 2013 :: B.19 Ingkarni Wardli :: Konrad Pilch :: University of Adelaide
Media...The KroneckerWeber 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. 

The Mathematics of Secrets 14:10 Mon 8 Apr, 2013 :: 210 Napier Building :: Dr Naomi Benger :: School of Mathematical Sciences
Media...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. 

A glimpse at the Langlands program 15:10 Fri 12 Apr, 2013 :: B.18 Ingkarni Wardli :: Dr Masoud Kamgarpour :: University of Queensland
Media...Abstract: In the late 1960s, Robert Langlands made a series of surprising conjectures relating fundamental concepts from number theory, representation theory, and algebraic geometry. Langlands' conjectures soon developed into a highprofile international research program known as the Langlands program. Many fundamental problems, including the ShimuraTaniyamaWeil conjecture (partially settled by Andrew Wiles in his proof of the Fermat's Last Theorem), are particular cases of the Langlands program. In this talk, I will discuss some of the motivation and results in this program. 

A strong Oka principle for proper immersions of finitely connected planar domains into CxC* 12:10 Fri 31 May, 2013 :: Ingkarni Wardli B19 :: Dr Tyson Ritter :: University of Adelaide
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. In previous work we showed that, given a continuous map from X to the elliptic manifold CxC*, where X is a finitely connected planar domain without isolated boundary points, a stronger Oka property holds whereby the map is homotopic to a proper holomorphic embedding. If the planar domain is additionally permitted to have isolated boundary points the problem becomes more difficult, and it is not yet clear whether a strong Oka property for embeddings into CxC* continues to hold. We will discuss recent results showing that every continuous map from a finitely connected planar domain into CxC* is homotopic to a proper immersion that, in most cases, identifies at most finitely many pairs of distinct points. This is joint work with Finnur Larusson. 

Invariant Theory: The 19th Century and Beyond 15:10 Fri 21 Jun, 2013 :: B.18 Ingkarni Wardli :: Dr Jarod Alper :: Australian National University
Media...A central theme in 19th century mathematics was invariant theory, which was viewed as a bridge between geometry and algebra. David Hilbert revolutionized the field with two seminal papers in 1890 and 1893 with techniques such as Hilbert's basis theorem, Hilbert's Nullstellensatz and Hilbert's syzygy theorem that spawned the modern field of commutative algebra. After Hilbert's groundbreaking work, the field of invariant theory remained largely inactive until the 1960's when David Mumford revitalized the field by reinterpreting Hilbert's ideas in the context of algebraic geometry which ultimately led to the influential construction of the moduli space of smooth curves. Today invariant theory remains a vital research area with connections to various mathematical disciplines: representation theory, algebraic geometry, commutative algebra, combinatorics and nonlinear differential operators.
The goal of this talk is to provide an introduction to invariant theory with an emphasis on Hilbert's and Mumford's contributions. Time permitting, I will explain recent research with Maksym Fedorchuk and David Smyth which exploits the ideas of Hilbert, Mumford as well as Kempf to answer a classical question concerning the stability of algebraic curves. 

The LowenheimSkolem theorem 12:10 Mon 26 Aug, 2013 :: B.19 Ingkarni Wardli :: William Crawford :: University of Adelaide
Media...For those of us who didn't do an undergrad course in logic, the foundations of set theory are pretty daunting. I will give a run down of some of the basics and then talk about a lesser known, but interesting result; the LowenheimSkolem theorem. One of the consequences of the theorem is that a set can be countable in one model of set theory, while being uncountable in another. 

Geometry of moduli spaces 12:10 Fri 30 Aug, 2013 :: Ingkarni Wardli B19 :: Prof Georg Schumacher :: University of Marburg
We discuss the concept of moduli spaces in complex geometry. The main examples are moduli of compact Riemann surfaces, moduli of compact projective varieties and moduli of holomorphic vector bundles, whose points correspond to isomorphism classes of the given objects. Moduli spaces carry a natural topology, whereas a complex structure that reflects the variation of the structure in a family exists in general only under extra conditions. In a similar way, a natural hermitian metric (WeilPetersson metric) on moduli spaces that induces a symplectic structure can be constructed from the variation of distinguished metrics on the fibers. In this way, various questions concerning the underlying symplectic structure, the curvature of the WeilPetersson metric, hyperbolicity of moduli spaces, and construction of positive/ample line bundles on compactified moduli spaces can be answered. 

The logarithmic singularities of the Green functions of the conformal powers of the Laplacian 11:10 Mon 16 Sep, 2013 :: Ingkarni Wardli B20 :: Prof Raphael Ponge :: Seoul National University
Green functions play an important role in conformal geometry. In this talk, we shall explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators are the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for PoincareEinstein metrics. The results are formulated in terms of Weyl conformal invariants defined via the ambient metric of FeffermanGraham. 

Noncommutative geometry and conformal geometry 13:10 Mon 16 Sep, 2013 :: Ingkarni Wardli B20 :: Prof Raphael Ponge :: Seoul National University
In this talk we shall report on a program of using the recent framework of twisted spectral triples to study conformal geometry from a noncommutative geometric perspective. One result is a local index formula in conformal geometry taking into account the action of the group of conformal diffeomorphisms. Another result is a version of VafaWitten's inequality for twisted spectral triples. Geometric applications include a version of VafaWitten's inequality in conformal geometry. There are also noncommutative versions for spectral triples over noncommutative tori and duals of discrete cocompact subgroups of semisimple Lie groups satisfying the BaumConnes conjecture. (This is joint work with Hang Wang.) 

Symmetry gaps for geometric structures 15:10 Fri 20 Sep, 2013 :: B.18 Ingkarni Wardli :: Dr Dennis The :: Australian National University
Media...Klein's Erlangen program classified geometries based on their (transitive) groups of symmetries, e.g. Euclidean geometry is the quotient of the rigid motion group by the subgroup of rotations. While this perspective is homogeneous, Riemann's generalization of Euclidean geometry is in general very "lumpy"  i.e. there exist Riemannian manifolds that have no symmetries at all. A common generalization where a group still plays a dominant role is Cartan geometry, which first arose in Cartan's solution to the equivalence problem for geometric structures, and which articulates what a "curved version" of a flat (homogeneous) model means. Parabolic geometries are Cartan geometries modelled on (generalized) flag varieties (e.g. projective space, isotropic Grassmannians) which are wellknown objects from the representation theory of semisimple Lie groups. These curved versions encompass a zoo of interesting geometries, including conformal, projective, CR, systems of 2nd order ODE, etc. This interaction between differential geometry and representation theory has proved extremely fruitful in recent years. My talk will be an examplebased tour of various types of parabolic geometries, which I'll use to outline some of the main aspects of the theory (suppressing technical details). The main thread throughout the talk will be the symmetry gap problem: For a given type of Cartan geometry, the maximal symmetry dimension is realized by the flat model, but what is the next possible ("submaximal") symmetry dimension? I'll sketch a recent solution (in joint work with Boris Kruglikov) for a wide class of parabolic geometries which gives a combinatorial recipe for reading the submaximal symmetry dimension from a Dynkin diagram. 

Localised index and L^2Lefschetz fixed point formula 12:10 Fri 25 Oct, 2013 :: Ingkarni Wardli B19 :: Dr Hang Wang :: University of Adelaide
In this talk we introduce a class of localised indices for the Dirac type operators on a complete Riemannian manifold, where a discrete group acts properly, cocompactly and isometrically. These localised indices, generalising the L^2index of Atiyah, are obtained by taking HattoriStallings traces of the higher index for the Dirac type operators. We shall talk about some motivation and applications for working on localised indices. The talk is related to joint work with BaiLing Wang. 

The density property for complex manifolds: a strong form of holomorphic flexibility 12:10 Fri 24 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Frank Kutzschebauch :: University of Bern
Compared with the real differentiable case, complex manifolds in general are more rigid, their groups of holomorphic diffeomorphisms are rather small (in general trivial). A long known exception to this behavior is affine nspace C^n for n at least 2. Its group of holomorphic diffeomorphisms is infinite dimensional. In the late 1980s Andersen and Lempert proved a remarkable
theorem which stated in its generalized version due to Forstneric and Rosay that any local holomorphic phase flow given on a Runge subset of C^n can be locally uniformly approximated by a global holomorphic diffeomorphism. The main ingredient in the proof was formalized by Varolin and called the density property: The Lie algebra generated by complete holomorphic vector fields is dense in the Lie algebra of all holomorphic vector fields. In these manifolds a similar local to global approximation of AndersenLempert type holds. It is a precise way of saying that the group of holomorphic diffeomorphisms is large.
In the talk we will explain how this notion is related to other more recent flexibility notions in complex geometry, in particular to the notion of a OkaForstneric manifold. We will give examples of manifolds with the density property and sketch applications of the density property. If time permits we will explain criteria for the density property developed by Kaliman and the speaker.


Integrability of infinitedimensional Lie algebras and Lie algebroids 12:10 Fri 7 Feb, 2014 :: Ingkarni Wardli B20 :: Christoph Wockel :: Hamburg University
Lie's Third Theorem states that each finitedimensional Lie algebra is the Lie algebra of a Lie group (we also say "integrates to a Lie group"). The corresponding statement for infinitedimensional Lie algebras or Lie algebroids is false and we will explain geometrically why this is the case. The underlying pattern is that of integration of central extensions of Lie algebras and Lie algebroids. This also occurs in other contexts, and we will explain some aspects of string group models in these terms. In the end we will sketch how the nonintegrability of Lie algebras and Lie algebroids can be overcome by passing to higher categorical objects (such as smooth stacks) and give a panoramic (but still conjectural) perspective on the precise relation of the various integrability problems.


Hormander's estimate, some generalizations and new applications 12:10 Mon 17 Feb, 2014 :: Ingkarni Wardli B20 :: Prof Zbigniew Blocki :: Jagiellonian University
Lars Hormander proved his estimate for the dbar equation in 1965. It is one the most important results in several complex variables (SCV). New applications have
emerged recently, outside of SCV. We will present three of them: the OhsawaTakegoshi extension theorem with optimal constant, the onedimensional Suita Conjecture, and Nazarov's approach to the BourgainMilman inequality from convex analysis. 

The structuring role of chaotic stirring on pelagic ecosystems 11:10 Fri 28 Feb, 2014 :: B19 Ingkarni Wardli :: Dr Francesco d'Ovidio :: Universite Pierre et Marie Curie (Paris VI)
The open ocean upper layer is characterized by a complex transport dynamics occuring over different spatiotemporal scales. At the scale of 10100 km  which covers the so called mesoscale and part of the submesoscale  in situ and remote sensing observations detect strong variability in physical and biogeochemical fields like sea surface temperature, salinity, and chlorophyll concentration. The calculation of Lyapunov exponent and other nonlinear diagnostics applied to the surface currents have allowed to show that an important part of this tracer variability is due to chaotic stirring. Here I will extend this analysis to marine ecosystems. For primary producers, I will show that stable and unstable manifolds of hyperbolic points embedded in the surface velocity field are able to structure the phytoplanktonic community in fluid dynamical niches of dominant types, where competition can locally occur during bloom events. By using data from tagged whales, frigatebirds, and elephant seals, I will also show that chaotic stirring affects the behaviour of higher trophic levels. In perspective, these relations between transport structures and marine ecosystems can be the base for a biodiversity index constructued from satellite information, and therefore able to monitor key aspects of the marine biodiversity and its temporal variability at the global scale. 

Moduli spaces of contact instantons 12:10 Fri 28 Mar, 2014 :: Ingkarni Wardli B20 :: David Baraglia :: University of Adelaide
In dimensions greater than four there are several notions of higher YangMills instantons. This talk concerns one such case, contact instantons, defined for 5dimensional contact manifolds. The geometry transverse to the Reeb foliation turns out to be important in understanding the moduli space. For example, we show the dimension of the moduli space is the index of a transverse elliptic complex. This is joint work with Pedram Hekmati. 

CARRYING CAPACITY FOR FINFISH AQUACULTURE IN SPENCER GULF: RAPID ASSESSMENT USING HYDRODYNAMIC AND NEARFIELD, SEMI  ANALYTIC SOLUTIONS 15:10 Fri 11 Apr, 2014 :: 5.58 Ingkarni Wardli :: Associate Professor John Middleton :: SARDI Aquatic Sciences and University of Adelaide
Aquaculture farming involves daily feeding of finfish and a subsequent excretion of nutrients into Spencer Gulf. Typically, finfish farming is done in six or so 50m diameter cages and over 600m X 600m lease sites. To help regulate the industry, it is desired that the finfish feed rates and the associated nutrient flux into the ocean are determined such that the maximum nutrient concentration c does not exceed a prescribed value (say cP) for ecosystem health. The prescribed value cP is determined by guidelines from the E.P.A. The concept is known as carrying capacity since limiting the feed rates limits the biomass of the farmed finfish.
Here, we model the concentrations that arise from a constant input flux (F) of nutrients in a source region (the cage or lease) using the (depthaveraged) two dimensional, advection diffusion equation for constant and sinusoidal (tides) currents. Application of the divergence theorem to this equation results in a new scale estimate of the maximum flux F (and thus feed rate) that is given by
F= cP /T* (1)
where cP is the maximum allowed concentration and T* is a new time scale of âflushingâ that involves both advection and diffusion. The scale estimate (1) is then shown to compare favourably with mathematically exact solutions of the advection diffusion equation that are obtained using Greenâs functions and Fourier transforms. The maximum nutrient flux and associated feed rates are then estimated everywhere in Spencer Gulf through the development and validation of a hydrodynamic model. The model provides seasonal averages of the mean currents U and horizontal diffusivities KS that are needed to estimate T*. The diffusivities are estimated from a shear dispersal model of the tides which are very large in the gulf. The estimates have been provided to PIRSA Fisheries and Aquaculture to assist in the sustainable expansion of finfish aquaculture.


Lefschetz fixed point theorem and beyond 12:10 Fri 2 May, 2014 :: Ingkarni Wardli B20 :: Hang Wang :: University of Adelaide
A Lefschetz number associated to a continuous map on a closed manifold is a topological invariant determined by the geometric information near the neighbourhood of fixed point set of the map. After an introduction of the Lefschetz fixed point theorem, we shall use the Diracdual Dirac method to derive the Lefschetz number on Ktheory level. The method concerns the comparison of the Dirac operator on the manifold and the Dirac operator on some submanifold. This method can be generalised to several interesting situations when the manifold is not necessarily compact. 

Ergodicity and loss of capacity: a stochastic horseshoe? 15:10 Fri 9 May, 2014 :: B.21 Ingkarni Wardli :: Professor Ami Radunskaya :: Pomona College, the United States of America
Media...Random fluctuations of an environment are common in ecological and
economical settings. The resulting processes can be described by a
stochastic dynamical system, where a family of maps parametrized by an
independent, identically distributed random variable forms the basis for a
Markov chain on a continuous state space. Random dynamical systems are a
beautiful combination of deterministic and random processes, and they have
received considerable interest since von Neuman and Ulam's seminal work in
the 1940's. Key questions in the study of a stochastic dynamical system
are: does the system have a welldefined average, i.e. is it ergodic?
How does this longterm behavior compare to that of the state
variable in a constant environment with the averaged parameter?
In this talk we answer these questions for a family of maps on the unit
interval that model selflimiting growth. The techniques used can be
extended to study other families of concave maps, and so we conjecture the
existence of a "stochastic horseshoe". 

The BismutChern character as dimension reduction functor and its twisting 12:10 Fri 4 Jul, 2014 :: Ingkarni Wardli B20 :: Fei Han :: National University of Singapore
The BismutChern character is a loop space refinement of the Chern character. It plays an essential role in the interpretation of the AtiyahSinger index theorem from the point of view of loop space. In this talk, I will first briefly review the construction of the BismutChern character and show how it can be viewed as a dimension reduction functor in the StolzTeichner program on supersymmetric quantum field theories. I will then introduce the construction of the twisted BismutChern character, which represents our joint work with Varghese Mathai. 

The Dirichlet problem for the prescribed Ricci curvature equation 12:10 Fri 15 Aug, 2014 :: Ingkarni Wardli B20 :: Artem Pulemotov :: University of Queensland
We will discuss the following question: is it possible to find a Riemannian metric whose Ricci curvature
is equal to a given tensor on a manifold M? To answer this question, one must analyze a weakly elliptic
secondorder geometric PDE. In the first part of the talk, we will review the history of the subject and state
several classical theorems. After that, our focus will be on new results concerning the case where M has
nonempty boundary. 

⌊ n!/e ⌉ 14:10 Tue 9 Sep, 2014 :: Ingkarni Wardli 715 Conference Room :: Dr. David Butler :: Maths Learning Centre
Media...What is this formula? Why does it use those strangely mismatched brackets, and why does it use both factorial and the number e? What is it supposed to calculate? And why would someone love it so much that they put it on a tshirt? In this seminar you will find out the answers to all of these questions, and also find out what derangements have to do with Taylor's theorem. 

Spectral asymptotics on random Sierpinski gaskets 12:10 Fri 26 Sep, 2014 :: Ingkarni Wardli B20 :: Uta Freiberg :: Universitaet Stuttgart
Self similar fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two model classes with the help of so called Vvariable fractals. This concept (developed by Barnsley, Hutchinson & Stenflo) allows the definition of new families of random fractals, hereby the parameter V describes the degree of `variability' of the realizations. We discuss how the degree of variability influences the geometric, analytic and stochastic properties of these sets.  These results have been obtained with Ben Hambly (University of Oxford) and John Hutchinson (ANU Canberra). 

Exploration vs. Exploitation with Partially Observable Gaussian Autoregressive Arms 15:00 Mon 29 Sep, 2014 :: Engineering North N132 :: Julia Kuhn :: The University of Queensland & The University of Amsterdam
Media...We consider a restless bandit problem with Gaussian autoregressive arms, where the state of an arm is only observed when it is played and the statedependent reward is collected. Since arms are only partially observable, a good decision policy needs to account for the fact that information about the state of an arm becomes more and more obsolete while the arm is not being played. Thus, the decision maker faces a tradeoff between exploiting those arms that are believed to be currently the most rewarding (i.e. those with the largest conditional mean), and exploring arms with a high conditional variance. Moreover, one would like the decision policy to remain tractable despite the infinite state space and also in systems with many arms. A policy that gives some priority to exploration is the Whittle index policy, for which we establish structural properties. These motivate a parametric index policy that is computationally much simpler than the Whittle index but can still outperform the myopic policy. Furthermore, we examine the manyarm behavior of the system under the parametric policy, identifying equations describing its asymptotic dynamics. Based on these insights we provide a simple heuristic algorithm to evaluate the performance of index policies; the latter is used to optimize the parametric index. 

The SerreGrothendieck theorem by geometric means 12:10 Fri 24 Oct, 2014 :: Ingkarni Wardli B20 :: David Roberts :: University of Adelaide
The SerreGrothendieck theorem implies that every torsion
integral 3rd cohomology class on a finite CWcomplex is the invariant
of some projective bundle. It was originally proved in a letter by
Serre, used homotopical methods, most notably a Postnikov
decomposition of a certain classifying space with divisible homotopy
groups. In this talk I will outline, using work of the algebraic
geometer Offer Gabber, a proof for compact smooth manifolds using
geometric means and a little Ktheory. 

Extending holomorphic maps from Stein manifolds into affine toric varieties 12:10 Fri 14 Nov, 2014 :: Ingkarni Wardli B20 :: Richard Larkang :: University of Adelaide
One way of defining socalled Oka manifolds is by saying that they satisfy the following interpolation property (IP): Y satisfies the IP if any holomorphic map from a closed submanifold S of a Stein manifold X into Y which has a continuous extension to X also has a holomorphic extension. An ostensibly weaker property is the convex interpolation property (CIP), where S is assumed to be a contractible submanifold of X = C^n. By a deep theorem of Forstneric, these (and several other) properties are in fact equivalent.
I will discuss a joint work with Finnur Larusson, where we consider the interpolation property when the target Y is a singular affine toric variety. We show that all affine toric varieties satisfy an interpolation property stronger than CIP, but that only in very special situations do they satisfy the full IP. 

Nonlinear analysis over infinite dimensional spaces and its applications 12:10 Fri 6 Feb, 2015 :: Ingkarni Wardli B20 :: Tsuyoshi Kato :: Kyoto University
In this talk we develop moduli theory of holomorphic curves over
infinite dimensional manifolds consisted by sequences of almost Kaehler manifolds.
Under the assumption of high symmetry, we verify that many mechanisms of
the standard moduli theory over closed symplectic manifolds also work over these
infinite dimensional spaces.
As an application, we study deformation theory of discrete groups acting
on trees. There is a canonical way, up to conjugacy to embed such groups
into the automorphism group over the infinite projective space.
We verify that for some class of Hamiltonian functions,
the deformed groups must be always asymptotically infinite. 

Indefinite spectral triples and foliations of spacetime 12:10 Fri 8 May, 2015 :: Napier 144 :: Koen van den Dungen :: Australian National University
Motivated by Dirac operators on Lorentzian manifolds, we propose a new framework to deal with nonsymmetric and nonelliptic operators in noncommutative geometry. We provide a definition for indefinite spectral triples, which correspond bijectively with certain pairs of spectral triples.
Next, we will show how a special case of indefinite spectral triples can be constructed from a family of spectral triples. In particular, this construction provides a convenient setting to study the Dirac operator on a spacetime with a foliation by spacelike hypersurfaces.
This talk is based on joint work with Adam Rennie (arXiv:1503.06916). 

Big things are weird 12:10 Mon 25 May, 2015 :: Napier LG29 :: Luke KeatingHughes :: University of Adelaide
Media...The pyramids of Giza, the depths of the Mariana trench, the massive Einstein Cross Quasar; all of these things are big and weird. Big weird things aren't just apparent in the physical world though, they appear in mathematics too! In this talk I will try to motivate a mathematical big thing and then show that it is weird.
In particular, we will introduce the necessary topology and homotopy theory in order to show that although all finite dimensional spheres are (almost canonically) noncontractible spaces  an infinite dimensional sphere IS contractible! This result's significance will then be explained in the context of Kuiper's Theorem if time permits. 

Dirac operators and Hamiltonian loop group action 12:10 Fri 24 Jul, 2015 :: Engineering and Maths EM212 :: Yanli Song :: University of Toronto
A definition to the geometric quantization for compact Hamiltonian Gspaces is given by Bott, defined as the index of the SpincDirac operator on the manifold. In this talk, I will explain how to generalize this idea to the Hamiltonian LGspaces. Instead of quantizing infinitedimensional manifolds directly, we use its equivalent finitedimensional model, the quasiHamiltonian Gspaces. By constructing twisted spinor bundle and twisted prequantum bundle on the quasiHamiltonian Gspace, we define a Dirac operator whose index are given by positive energy representation of loop groups. A key role in the construction will be played by the algebraic cubic Dirac operator for loop algebra. If time permitted, I will also explain how to prove the quantization commutes with reduction theorem for Hamiltonian LGspaces under this framework. 

Workshop on Geometric Quantisation 10:10 Mon 27 Jul, 2015 :: Level 7 conference room Ingkarni Wardli :: Michele Vergne, Weiping Zhang, Eckhard Meinrenken, Nigel Higson and many others
Media...Geometric quantisation has been an increasingly active area since before the 1980s, with links to physics, symplectic geometry, representation theory, index theory, and differential geometry and geometric analysis in general. In addition to its relevance as a field on its own, it acts as a focal point for the interaction between all of these areas, which has yielded farreaching and powerful results. This workshop features a large number of international speakers, who are all wellknown for their work in (differential) geometry, representation theory and/or geometric analysis. This is a great opportunity for anyone interested in these areas to meet and learn from some of the top mathematicians in the world. Students are especially welcome. Registration is free. 

Quantising proper actions on Spinc manifolds 11:00 Fri 31 Jul, 2015 :: Ingkarni Wardli Level 7 Room 7.15 :: Peter Hochs :: The University of Adelaide
Media...For a proper action by a Lie group on a Spinc manifold (both of which may be noncompact), we study an index of deformations of the Spinc Dirac operator, acting on the space of spinors invariant under the group action. When applied to spinors that are square integrable transversally to orbits in a suitable sense, the kernel of this operator turns out to be finitedimensional, under certain hypotheses of the deformation. This also allows one to show that the index has the quantisation commutes with reduction property (as proved by Meinrenken in the compact symplectic case, and by ParadanVergne in the compact Spinc case), for sufficiently large powers of the determinant line bundle. Furthermore, this result extends to Spinc Dirac operators twisted by vector bundles. A key ingredient of the arguments is the use of a family of inner products on the Lie algebra, depending on a point in the manifold. This is joint work with Mathai Varghese. 

Deformation retractions from the space of continuous maps between domains in C onto the space of holomorphic maps 12:10 Mon 17 Aug, 2015 :: Benham Labs G10 :: Brett Chenoweth :: University of Adelaide
Media...Mikhail Gromov proved in 1989 that every continuous map from a Stein manifold S to an elliptic manifold X could be deformed to a holomorphic map. More generally, it is true that if X is an Oka manifold then a continuous map from a Stein source into X can always be deformed to a holomorphic map. The question is whether we can do this for all continuous maps at once, in a `nice' way that does not change a map f if f is already holomorphic. In a recent paper by Larusson, we see that ANRs play an important in producing a partial answer to this question. In this talk we will explore the question in the relatively simple situation where the source and target are domains in the complex plane. 

Integrability conditions for the Grushin operators 12:10 Fri 4 Sep, 2015 :: Ingkarni Wardli B17 :: Michael Eastwood :: The University of Adelaide
Fix a nonnegative integer k and consider the vector fields in the plane X=d/dx and Y=x^kd/dy. A smooth function f(x,y) is locally constant if and only if it is annihilated by the k^th Grushin operator f\mapsto(Xf,Yf). What about the range of this operator?


Bezout's Theorem 12:10 Mon 7 Sep, 2015 :: Benham Labs G10 :: David Bowman :: University of Adelaide
Media...Generically, a line intersects a parabola at two distinct points. BezoutÃ¢ÂÂs theorem generalises this idea to the intersection of two arbitrary polynomial plane curves. We discuss exceptional cases and how they are corrected by introducing the notion of multiplicity and by extending the plane to projective space. We shall also discuss applications, time permitting.


Analytic complexity of bivariate holomorphic functions and cluster trees 12:10 Fri 2 Oct, 2015 :: Ingkarni Wardli B17 :: Timur Sadykov :: Plekhanov University, Moscow
The KolmogorovArnold theorem yields a representation of a multivariate continuous function in terms of a composition of functions which depend on at most two variables. In the analytic case, understanding the complexity of such a representation naturally leads to the notion of the analytic complexity of (a germ of) a bivariate multivalued analytic function. According to Beloshapka's local definition, the order of complexity of any univariate function is equal to zero while the nth complexity class is defined recursively to consist of functions of the form a(b(x,y)+c(x,y)), where a is a univariate analytic function and b and c belong to the (n1)th complexity class. Such a represenation is meant to be valid for suitable germs of multivalued holomorphic functions.
A randomly chosen bivariate analytic functions will most likely have infinite analytic complexity. However, for a number of important families of special functions of mathematical physics their complexity is finite and can be computed or estimated. Using this, we introduce the notion of the analytic complexity of a binary tree, in particular, a cluster tree, and investigate its properties.


Real Lie Groups and Complex Flag Manifolds 12:10 Fri 9 Oct, 2015 :: Ingkarni Wardli B17 :: Joseph A. Wolf :: University of California, Berkeley
Media...Let G be a complex simple direct limit group. Let G_R be a real form of G that corresponds to an hermitian symmetric space. I'll describe the corresponding bounded symmetric domain in the context of the Borel embedding, Cayley transforms, and the BergmanShilov boundary. Let Q be a parabolic subgroup of G. In finite dimensions this means that G/Q is a complex projective variety, or equivalently has a Kaehler metric invariant under a maximal compact subgroup of G. Then I'll show just how the bounded symmetric domains describe cycle spaces for open G_R orbits on G/Q. These cycle spaces include the complex bounded symmetric domains. In finite dimensions they are tightly related to moduli spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite dimensions there are more problems than answers. Finally, time permitting, I'll indicate how some of this goes over to real and to quaternionic bounded symmetric domains.


ChernSimons classes on loop spaces and diffeomorphism groups 12:10 Fri 16 Oct, 2015 :: Ingkarni Wardli B17 :: Steve Rosenberg :: Boston University
Media...Not much is known about the topology of the diffeomorphism group Diff(M) of manifolds M of dimension four and higher. We'll show that for a class of manifolds of dimension 4k+1, Diff(M) has infinite fundamental group. This is proved by translating the problem into a question about ChernSimons classes on the tangent bundle to the loop space LM. To build the CS classes, we use a family of metrics on LM associated to a Riemannian metric on M. The curvature of these metrics takes values in an algebra of pseudodifferential operators. The main technical step in the CS construction is to replace the ordinary matrix trace in finite dimensions with the Wodzicki residue, the unique trace on this algebra. The moral is that some techniques in finite dimensional Riemannian geometry can be extended to some examples in infinite dimensional geometry.


A fixed point theorem on noncompact manifolds 12:10 Fri 12 Feb, 2016 :: Ingkarni Wardli B21 :: Peter Hochs :: University of Adelaide / Radboud University
Media...For an elliptic operator on a compact manifold acted on by a compact Lie group, the AtiyahSegalSinger fixed point formula expresses its equivariant index in terms of data on fixed point sets of group elements. This can for example be used to prove Weylâs character formula. We extend the definition of the equivariant index to noncompact manifolds, and prove a generalisation of the AtiyahSegalSinger formula, for group elements with compact fixed point sets. In one example, this leads to a relation with characters of discrete series representations of semisimple Lie groups. (This is joint work with Hang Wang.) 

Tduality for elliptic curve orientifolds 12:10 Fri 4 Mar, 2016 :: Ingkarni Wardli B17 :: Jonathan Rosenberg :: University of Maryland
Media...Orientifold string theories are quantum field theories based on the
geometry of a space with an involution. Tdualities are certain
relationships between such theories that look different
on the surface but give rise to the same observable physics.
In this talk I will not assume
any knowledge of physics but will concentrate on the associated
geometry, in the case where the underlying space is a (complex)
elliptic curve and the involution is either holomorphic or
antiholomorphic. The results blend algebraic topology
and algebraic geometry. This is mostly joint work with
Chuck Doran and Stefan MendezDiez. 

Geometric analysis of gaplabelling 12:10 Fri 8 Apr, 2016 :: Eng & Maths EM205 :: Mathai Varghese :: University of Adelaide
Media...Using an earlier result, joint with Quillen, I will formulate a gap labelling conjecture for magnetic Schrodinger operators with smooth aperiodic potentials on Euclidean space. Results in low dimensions will be given, and the formulation of the same problem for certain nonEuclidean spaces will be given if time permits.
This is ongoing joint work with Moulay Benameur.


Sard Theorem for the endpoint map in subRiemannian manifolds 12:10 Fri 29 Apr, 2016 :: Eng & Maths EM205 :: Alessandro Ottazzi :: University of New South Wales
Media...SubRiemannian geometries occur in several areas of pure and applied mathematics, including harmonic analysis, PDEs, control theory, metric geometry, geometric group theory, and neurobiology. We introduce subRiemannian manifolds and give some examples. Therefore we discuss some of the open problems, and in particular we focus on the Sard Theorem for the endpoint map, which is related to the study of length minimizers. Finally, we consider some recent results obtained in collaboration with E. Le Donne, R. Montgomery, P. Pansu and D. Vittone. 

How to count Betti numbers 12:10 Fri 6 May, 2016 :: Eng & Maths EM205 :: David Baraglia :: University of Adelaide
Media...I will begin this talk by showing how to obtain the Betti numbers of certain smooth complex projective varieties by counting points over a finite field. For singular or noncompact varieties this motivates us to consider the "virtual Hodge numbers" encoded by the "HodgeDeligne polynomial", a refinement of the topological Euler characteristic. I will then discuss the computation of HodgeDeligne polynomials for certain singular character varieties (i.e. moduli spaces of flat connections). 

Harmonic analysis of HodgeDirac operators 12:10 Fri 13 May, 2016 :: Eng & Maths EM205 :: Pierre Portal :: Australian National University
Media...When the metric on a Riemannian manifold is perturbed in a rough (merely bounded and measurable) manner, do basic estimates involving the Hodge Dirac operator $D = d+d^*$ remain valid? Even in the model case of a perturbation of the euclidean metric on $\mathbb{R}^n$, this is a difficult question. For instance, the fact that the $L^2$ estimate $\Du\_2 \sim \\sqrt{D^{2}}u\_2$ remains valid for perturbed versions of $D$ was a famous conjecture made by Kato in 1961 and solved, positively, in a ground breaking paper of Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002. In the past fifteen years, a theory has emerged from the solution of this conjecture, making rough perturbation problems much more tractable. In this talk, I will give a general introduction to this theory, and present one of its latest results: a flexible approach to $L^p$ estimates for the holomorphic functional calculus of $D$. This is joint work with D. Frey (Delft) and A. McIntosh (ANU).


On the Strong Novikov Conjecture for Locally Compact Groups in Low Degree Cohomology Classes 12:10 Fri 3 Jun, 2016 :: Eng & Maths EM205 :: Yoshiyasu Fukumoto :: Kyoto University
Media...The main result I will discuss is nonvanishing of the image of the index map from the Gequivariant Khomology of a Gmanifold X to the Ktheory of the C*algebra of the group G. The action of G on X is assumed to be proper and cocompact. Under the assumption that the Kronecker pairing of a Khomology class with a lowdimensional cohomology class is nonzero, we prove that the image of this class under the index map is nonzero. Neither discreteness of the locally compact group G nor freeness of the action of G on X are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.


Holomorphic Flexibility Properties of Spaces of Elliptic Functions 12:10 Fri 29 Jul, 2016 :: Ingkarni Wardli B18 :: David Bowman :: University of Adelaide
The set of meromorphic functions on an elliptic curve naturally possesses the structure of a complex manifold. The component of degree 3 functions is 6dimensional and enjoys several interesting complexanalytic properties that make it, loosely speaking, the opposite of a hyperbolic manifold. Our main result is that this component has a 54sheeted branched covering space that is an Oka manifold. 

Calculus on symplectic manifolds 12:10 Fri 12 Aug, 2016 :: Ingkarni Wardli B18 :: Mike Eastwood :: University of Adelaide
Media...One can use the symplectic form to construct an elliptic complex replacing the de Rham complex. Then, under suitable curvature conditions, one can form coupled versions of this complex. Finally, on complex projective space, these constructions give rise to a series of elliptic complexes with geometric consequences for the FubiniStudy metric and its Xray transform. This talk, which will start from scratch, is based on the work of many authors but, especially, current joint work with Jan Slovak. 

Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type 12:10 Fri 19 Aug, 2016 :: Ingkarni Wardli B18 :: Lesley Ward :: University of South Australia
Media...Much effort has been devoted to generalizing the
Calder'onZygmund theory in harmonic analysis from Euclidean
spaces to metric measure spaces, or spaces of homogeneous type.
Here the underlying space R^n with Euclidean metric
and Lebesgue measure is replaced by a set X with general
metric or quasimetric and a doubling measure. Further, one can
replace the Laplacian operator that underpins the
CalderonZygmund theory by more general operators L
satisfying heat kernel estimates.
I will present recent joint work with P. Chen, X.T. Duong,
J. Li and L.X. Yan along these lines. We develop the theory of
product Hardy spaces H^p_{L_1,L_2}(X_1 x X_2), for 1 

What is the best way to count votes? 13:10 Mon 12 Sep, 2016 :: Hughes 322 :: Dr Stuart Johnson :: School of Mathematical Sciences
Media...Around the world there are many different ways of counting votes in elections, and even within Australia there are different methods in use in various states. Which is the best method? Even for the simplest case of electing one person in a single electorate there is no easy answer to this, in fact there is a famous result  Arrow's Theorem  which tells us that there is no perfect way of counting votes. I will describe a number of different methods along with their problems before giving a more precise statement of the theorem and outlining a proof 

Geometry of pseudodifferential algebra bundles 12:10 Fri 16 Sep, 2016 :: Ingkarni Wardli B18 :: Mathai Varghese :: University of Adelaide
Media...I will motivate the construction of pseudodifferential algebra bundles arising in index theory, and also outline the construction of general pseudodifferential algebra bundles (and the associated sphere bundles), showing that there are many that are purely infinite dimensional that do not come from usual constructions in index theory. I will also discuss characteristic classes of such bundles. This is joint work with Richard Melrose. 

Symmetric functions and quantum integrability 15:10 Fri 30 Sep, 2016 :: Napier G03 :: Dr Paul ZinnJustin :: University of Melbourne/Universite Pierre et Marie Curie
Media...We'll discuss an approach to studying families of symmetric polynomials which is based on ''quantum integrability'', that is, on the use of exactly solvable twodimensional lattice models. We'll first explain the general strategy on the simplest case, namely Schur polynomials, with the introduction of a model of lattice paths (a.k.a. fivevertex model). We'll then discuss recent work (in collaboration with M. Wheeler) that extends this approach to HallLittlewood polynomials and Grothendieck polynomials, and some applications of it. 

Character Formula for Discrete Series 12:10 Fri 14 Oct, 2016 :: Ingkarni Wardli B18 :: Hang Wang :: University of Adelaide
Media...Weyl character formula describes characters of irreducible representations of compact Lie groups. This formula can be obtained using geometric method, for example, from the AtiyahBott fixed point theorem or the AtiyahSegalSinger index theorem. HarishChandra character formula, the noncompact analogue of the Weyl character formula, can also be studied from the point of view of index theory. We apply orbital integrals on Ktheory of HarishChandra Schwartz algebra of a semisimple Lie group G, and then use geometric method to deduce HarishChandra character formulas for discrete series representations of G. This is work in progress with Peter Hochs.


What is index theory? 12:10 Tue 21 Mar, 2017 :: Inkgarni Wardli 5.57 :: Dr Peter Hochs :: School of Mathematical Sciences
Media...Index theory is a link between topology, geometry and analysis. A typical theorem in index theory says that two numbers are equal: an analytic index and a topological index. The first theorem of this kind was the index theorem of Atiyah and Singer, which they proved in 1963. Index theorems have many applications in maths and physics. For example, they can be used to prove that a differential equation must have a solution. Also, they imply that the topology of a space like a sphere or a torus determines in what ways it can be curved. Topology is the study of geometric properties that do not change if we stretch or compress a shape without cutting or glueing. Curvature does change when we stretch something out, so it is surprising that topology can say anything about curvature. Index theory has many surprising consequences like this.


Ktypes of tempered representations 12:10 Fri 7 Apr, 2017 :: Napier 209 :: Peter Hochs :: University of Adelaide
Media...Tempered representations of a reductive Lie group G are the irreducible unitary representations one needs in the Plancherel decomposition of L^2(G). They are relevant to harmonic analysis because of this, and also occur in the Langlands classification of the larger class of admissible representations. If K in G is a maximal compact subgroup, then there is a considerable amount of information in the restriction of a tempered representation to K. In joint work with Yanli Song and Shilin Yu, we give a geometric expression for the decomposition of such a restriction into irreducibles. The multiplicities of these irreducibles are expressed as indices of Dirac operators on reduced spaces of a coadjoint orbit of G corresponding to the representation. These reduced spaces are Spinc analogues of reduced spaces in symplectic geometry, defined in terms of moment maps that represent conserved quantities. This result involves a Spinc version of the quantisation commutes with reduction principle for noncompact manifolds. For discrete series representations, this was done by Paradan in 2003. 

Hodge theory on the moduli space of Riemann surfaces 12:10 Fri 5 May, 2017 :: Napier 209 :: Jesse GellRedman :: University of Melbourne
Media...The Hodge theorem on a closed Riemannian manifold identifies the deRham cohomology with the space of harmonic differential forms. Although there are various extensions of the Hodge theorem to singular or complete but noncompact spaces, when there is an identification of L^2 Harmonic forms with a topological feature of the underlying space, it is highly dependent on the nature of infinity (in the noncompact case) or the locus of incompleteness; no unifying theorem treats all cases. We will discuss work toward extending the Hodge theorem to singular Riemannian manifolds where the singular locus is an incomplete cusp edge. These can be pictured locally as a bundle of horns, and they provide a model for the behavior of the WeilPetersson metric on the compactified Riemann moduli space near the interior of a divisor. Joint with J. Swoboda and R. Melrose. 

Schubert Calculus on Lagrangian Grassmannians 12:10 Tue 23 May, 2017 :: EM 213 :: Hiep Tuan Dang :: National centre for theoretical sciences, Taiwan
Media...The Lagrangian Grassmannian $LG = LG(n,2n)$ is the projective complex manifold which parametrizes Lagrangian (i.e. maximal isotropic) subspaces in a symplective vector space of dimension $2n$. This talk is mainly devoted to Schubert calculus on $LG$. We first recall the definition of Schubert classes in this context. Then we present basic results which are similar to the classical formulas due to Pieri and Giambelli. These lead to a presentation of the cohomology ring of $LG$. Finally, we will discuss recent results related to the Schubert structure constants and GromovWitten invariants of $LG$. 

Compact pseudoRiemannian homogeneous spaces 12:10 Fri 18 Aug, 2017 :: Engineering Sth S111 :: Wolfgang Globke :: University of Adelaide
Media...A pseudoRiemannian homogeneous space $M$ of finite volume can be presented as $M=G/H$, where $G$ is a Lie group acting transitively and isometrically on $M$, and $H$ is a closed subgroup of $G$.
The condition that $G$ acts isometrically and thus preserves a finite measure on $M$ leads to strong algebraic restrictions on $G$. In the special case where $G$ has no compact semisimple normal subgroups, it turns out that the isotropy subgroup $H$ is a lattice, and that the metric on $M$ comes from a biinvariant metric on $G$.
This result allows us to recover Zeghibâs classification of Lorentzian compact homogeneous spaces, and to move towards a classification for metric index 2.
As an application we can investigate which pseudoRiemannian homogeneous spaces of finite volume are Einstein spaces. Through the existence questions for lattice subgroups, this leads to an interesting connection with the theory of transcendental numbers, which allows us to characterize the Einstein cases in low dimensions.
This talk is based on joint works with Oliver Baues, Yuri Nikolayevsky and Abdelghani Zeghib. 

On directions and operators 11:10 Wed 27 Sep, 2017 :: Engineering & Math EM213 :: Malabika Pramanik :: University of British Columbia
Media...Many fundamental operators arising in harmonic analysis are governed by sets of directions that they are naturally associated with. This talk will survey a few representative results in this area, and report on some new developments. 

Equivariant formality of homogeneous spaces 12:10 Fri 29 Sep, 2017 :: Engineering Sth S111 :: Alex ChiKwong Fok :: University of Adelaide
Equivariant formality, a notion in equivariant topology introduced by GoreskyKottwitzMacpherson, is a desirable property of spaces with group actions, which allows the application of localisation formula to evaluate integrals of any top closed forms and enables one to compute easily the equivariant cohomology. Broad classes of spaces of especial interest are wellknown to be equivariantly formal, e.g., compact symplectic manifolds equipped with Hamiltonian compact Lie group actions and projective varieties equipped with linear algebraic torus actions, of which flag varieties are examples. Less is known about compact homogeneous spaces G/K equipped with the isotropy action of K, which is not necessarily of maximal rank. In this talk we will review previous attempts of characterizing equivariant formality of G/K, and present our recent results on this problem using an analogue of equivariant formality in Ktheory. Part of the work presented in this talk is joint with Jeffrey Carlson. 

Endperiodic Khomology and spin bordism 12:10 Fri 20 Oct, 2017 :: Engineering Sth S111 :: Michael Hallam :: University of Adelaide
This talk introduces new "endperiodic" variants of geometric Khomology and spin bordism theories that are tailored to a recent index theorem for evendimensional manifolds with periodic ends. This index theorem, due to Mrowka, Ruberman and Saveliev, is a generalisation of the AtiyahPatodiSinger index theorem for manifolds with odddimensional boundary. As in the APS index theorem, there is an (endperiodic) eta invariant that appears as a correction term for the periodic end. Invariance properties of the standard relative eta invariants are elegantly expressed using Khomology and spin bordism, and this continues to hold in the endperiodic case. In fact, there are natural isomorphisms between the standard Khomology/bordism theories and their endperiodic versions, and moreover these isomorphisms preserve relative eta invariants. The study is motivated by results on positive scalar curvature, namely obstructions and distinct path components of the moduli space of PSC metrics. Our isomorphisms provide a systematic method for transferring certain results on PSC from the odddimensional case to the evendimensional case. This work is joint with Mathai Varghese. 

A Hecke module structure on the KKtheory of arithmetic groups 13:10 Fri 2 Mar, 2018 :: Barr Smith South Polygon Lecture theatre :: Bram Mesland :: University of Bonn
Media...Let $G$ be a locally compact group, $\Gamma$ a discrete subgroup and $C_{G}(\Gamma)$ the commensurator of $\Gamma$ in $G$. The cohomology of $\Gamma$ is a module over the Shimura Hecke ring of the pair $(\Gamma,C_G(\Gamma))$. This construction recovers the action of the Hecke operators on modular forms for $SL(2,\mathbb{Z})$ as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair $(\Gamma, C_{G}(\Gamma))$ maps into the $KK$ring associated to an arbitrary $\Gamma$C*algebra. From this we obtain a variety of $K$theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of $\Gamma$spaces. Examples include the BorelSerre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the BruhatTits tree of $SL(2,\mathbb{Z})$. This is joint work with M.H. Sengun (Sheffield). 

Radial Toeplitz operators on bounded symmetric domains 11:10 Fri 9 Mar, 2018 :: Lower Napier LG11 :: Raul QuirogaBarranco :: CIMAT, Guanajuato, Mexico
Media...The Bergman spaces on a complex domain are defined as the space of holomorphic squareintegrable functions on the domain. These carry interesting structures both for analysis and representation theory in the case of bounded symmetric domains. On the other hand, these spaces have some bounded operators obtained as the composition of a multiplier operator and a projection. These operators are highly noncommuting between each other. However, there exist large commutative C*algebras generated by some of these Toeplitz operators very much related to Lie groups. I will construct an example of such C*algebras and provide a fairly explicit simultaneous diagonalization of the generating Toeplitz operators. 

Quantum Airy structures and topological recursion 13:10 Wed 14 Mar, 2018 :: Ingkarni Wardli B17 :: Gaetan Borot :: MPI Bonn
Media...Quantum Airy structures are Lie algebras of quadratic differential operators  their classical limit describes Lagrangian subvarieties in symplectic vector spaces which are tangent to the zero section and cut out by quadratic equations. Their partition function  which is the function annihilated by the collection of differential operators  can be computed by the topological recursion. I will explain how to obtain quantum Airy structures from spectral curves, and explain how we can retrieve from them correlation functions of semisimple cohomological field theories, by exploiting the symmetries. This is based on joint work with Andersen, Chekhov and Orantin. 

Family gauge theory and characteristic classes of bundles of 4manifolds 13:10 Fri 16 Mar, 2018 :: Barr Smith South Polygon Lecture theatre :: Hokuto Konno :: University of Tokyo
Media...I will define a nontrivial characteristic class of bundles of
4manifolds using families of SeibergWitten equations. The basic idea
of the construction is to consider an infinite dimensional
analogue of the Euler class used in the usual theory of characteristic
classes. I will also explain how to prove the nontriviality of this
characteristic class. If time permits, I will mention a relation between
our characteristic class and positive scalar curvature metrics. 

Index of Equivariant CalliasType Operators 13:10 Fri 27 Apr, 2018 :: Barr Smith South Polygon Lecture theatre :: Hao Guo :: University of Adelaide
Media...Suppose M is a smooth Riemannian manifold on which a Lie group G acts properly and isometrically. In this talk I will explore properties of a particular class of Ginvariant operators on M, called GCalliastype operators. These are Dirac operators that have been given an additional Z_2grading and a perturbation so as to be "invertible outside of a cocompact set in M". It turns out that GCalliastype operators are equivariantly Fredholm and so have an index in the Ktheory of the maximal group C*algebra of G. This index can be expressed as a KKproduct of a class in Khomology and a class in the Ktheory of the Higson Gcorona. In fact, one can show that the Ktheory of the Higson Gcorona is highly nontrivial, and thus the index theory of GCalliastype operators is not obviously trivial. As an application of the index theory of GCalliastype operators, I will mention an obstruction to the existence of Ginvariant metrics of positive scalar curvature on M. 

Obstructions to smooth group actions on 4manifolds from families SeibergWitten theory 13:10 Fri 25 May, 2018 :: Barr Smith South Polygon Lecture theatre :: David Baraglia :: University of Adelaide
Media...Let X be a smooth, compact, oriented 4manifold and consider the following problem. Let G be a group which acts on the second cohomology of X preserving the intersection form. Can this action of G on H^2(X) be lifted to an action of G on X by diffeomorphisms? We study a parametrised version of SeibergWitten theory for smooth families of 4manifolds and obtain obstructions to the existence of such lifts. For example, we construct compact simplyconnected 4manifolds X and involutions on H^2(X) that can be realised by a continuous involution on X, or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on X. 

The mass of Riemannian manifolds 13:10 Fri 1 Jun, 2018 :: Barr Smith South Polygon Lecture theatre :: Matthias Ludewig :: MPIM Bonn
We will define the mass of differential operators L on compact Riemannian manifolds. In odd dimensions, if L is a conformally covariant differential operator, then its mass is also conformally covariant, while in even dimensions, one has a more complicated transformation rule. In the special case that L is the Yamabe operator, its mass is related to the ADM mass of an associated asymptotically flat spacetime. In particular, one expects positive mass theorems in various settings. Here we highlight some recent results. 

Carleman approximation of maps into Oka manifolds. 11:10 Fri 3 Aug, 2018 :: Barr Smith South Polygon Lecture theatre :: Brett Chenoweth :: University of Ljubljana
In 1927 Torsten Carleman proved a remarkable extension of the StoneWeierstrass theorem. Carlemanâs theorem is ostensibly the first result concerning the approximation of functions on unbounded closed subsets of C by entire functions. In this talk we introduce Carlemanâs theorem and several of its recent generalisations including the titled generalisation which was proved by the speaker in arXiv:1804.10680. 

Equivariant Index, Traces and Representation Theory 11:10 Fri 10 Aug, 2018 :: Barr Smith South Polygon Lecture theatre :: Hang Wang :: University of Adelaide
Ktheory of C*algebras associated to a semisimple Lie group can be understood both from the geometric point of view via BaumConnes assembly map and from the representation theoretic point of view via harmonic analysis of Lie groups. A Ktheory generator can be viewed as the equivariant index of some Dirac operator, but also interpreted as a (family of) representation(s) parametrised by the noncompact abelian part in the Levi component of a cuspidal parabolic subgroup. Applying orbital traces to the Ktheory group, we obtain the equivariant index as a fixed point formula which, for each Ktheory generators for (limit of) discrete series, recovers HarishChandraâs character formula on the representation theory side. This is a noncompact analogue of AtiyahSegalSinger fixed point theorem in relation to the Weyl character formula. This is joint work with Peter Hochs. 

Interactive theorem proving for mathematicians 15:10 Fri 5 Oct, 2018 :: Napier 208 :: A/Prof Scott Morrison :: Australian National University
Mathematicians use computers to write their proofs (LaTeX), and to do their calculations (Sage, Mathematica, Maple, Matlab, etc, as well as custom code for simulations or searches). However today we rarely use computers to help us to construct and understand proofs.
There is a long tradition in computer science of interactive and automatic theorem proving; particularly today these are important tools in engineering correct software, as well as in optimisation and compilation. There have been some notable examples of formalisation of modern mathematics (e.g. the odd order theorem, the Kepler conjecture, and the fourcolour theorem). Even in these cases, huge engineering efforts were required to translate the mathematics to a form a computer could understand. Moreover, in most areas of research there is a huge gap between the interests of human mathematicians and the abilities of computer provers.
Nevertheless, I think it's time for mathematicians to start getting interested in interactive theorem provers! It's now possible to write proofs, and write tools that help write proofs, in languages which are expressive enough to encompass most of modern mathematics, and ergonomic enough to use for general purpose programming.
I'll give an informal introduction to dependent type theory (the logical foundation of many modern theorem provers), some examples of doing mathematics in such a system, and my experiences working with mathematics students in these systems. 

Twisted Ktheory of compact Lie groups and extended Verlinde algebras 11:10 Fri 12 Oct, 2018 :: Barr Smith South Polygon Lecture theatre :: ChiKwong Fok :: University of Adelaide
In a series of recent papers, Freed, Hopkins and Teleman put forth a deep result which identifies the twisted K theory of a compact Lie group G with the representation theory of its loop group LG. Under suitable conditions, both objects can be enhanced to the Verlinde algebra, which appears in mathematical physics as the Frobenius algebra of a certain topological quantum field theory, and in algebraic geometry as the algebra encoding information of moduli spaces of Gbundles over Riemann surfaces. The Verlinde algebra for G with nice connectedness properties have been wellknown. However, explicit descriptions of such for disconnected G are lacking. In this talk, I will discuss the various aspects of the FreedHopkinsTeleman Theorem and partial results on an extension of the Verlinde algebra arising from a disconnected G. The talk is based on work in progress joint with David Baraglia and Varghese Mathai. 
News matching "The index theorem for projective families of ellip" 
Stoneham Prize The inaugural Stoneham Prize, awarded for the best poster by a graduate student in the first two years of their candidature, was awarded on the 4th of April. The winner was Ric Green, for his poster "What is Geometry?". Two Viewers' Choice prizes were also awarded to Ray Vozzo for his poster "The 7 Bridges of Koenigsberg  The 1st Theorem in Topology" and David Butler for his poster "The Queen of Hearts Plays Noughts and Crosses". Posted Sun 13 Apr 08. 

ARC Grant successes Congratulations to Tony Roberts, Charles Pearce, Robert Elliot, Andrew Metcalfe and all their collaborators on their success in the current round of ARC grants. The projects are "Development of innovative technologies for oil production based on the advanced theory of suspension flows in porous media" (Tony Roberts et al.), "Perturbation and approximation methods for linear operators with applications to train control, water resource management and evolution of physical systems" (Charles Pearce et al.),
"Risk Measures and Management in Finance and Actuarial Science Under RegimeSwitching Models" (Robert Elliott et al.) and "A new flood design methodology for a variable and changing climate" (Andrew Metcalfe et al.) Posted Mon 26 Oct 09. 

IGA Lecture Series by Professor Dan Freed The School of Mathematical Sciences will host a series of lectures by Professor Dan Freed (University of Texas, Austin) as part of an upcoming IGA/AMSI workshop, October 1822, 2010. Details of the workshop can be found here. Posted Tue 5 Oct 10. 

ARC Grant Success Congratulations to the following staff who were successful in securing funding from the Australian Research Council Discovery Projects Scheme. Associate Professor Finnur Larusson awarded $270,000 for his project Flexibility and symmetry in complex geometry; Dr Thomas Leistner, awarded $303,464 for his project Holonomy groups in Lorentzian geometry, Professor Michael Murray Murray and Dr Daniel Stevenson (Glasgow), awarded $270,000 for their project Bundle gerbes: generalisations and applications; Professor Mathai Varghese, awarded $105,000 for his project Advances in index theory and Prof Anthony Roberts and Professor Ioannis Kevrekidis (Princeton) awarded $330,000 for their project Accurate modelling of large multiscale dynamical systems for engineering and scientific
simulation and analysis Posted Tue 8 Nov 11. 

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

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

Inversion of analytically perturbed linear operators that are singular at the origin Howlett, P; Avrachenkov, K; Pearce, Charles; Ejov, V, Journal of Mathematical Analysis and Applications 353 (68–84) 2009  Unitals in projective planes Barwick, Susan; Ebert, G, (Springer) 2008  Metric connections in projective differential geometry Eastwood, Michael; Matveev, V, Symmetries and Overdetermined Systems of Partial Differential Equations, USA 17/07/08  Notes on projective differential geometry Eastwood, Michael, Symmetries and Overdetermined Systems of Partial Differential Equations, USA 17/07/08  Equivariant and fractional index of projective elliptic operators Varghese, Mathai; Melrose, R; Singer, I, Journal of Differential Geometry 78 (465–473) 2008  A note on Nk configurations and theorems in projective space Glynn, David, Bulletin of the Australian Mathematical Society 76 (15–31) 2007  A sequence approach to linear perfect hash families Barwick, Susan; Jackson, WenAi, Designs Codes and Cryptography 45 (95–121) 2007  Aspects of Dirac operators in analysis Eastwood, Michael; Ryan, J, Milan Journal of Mathematics 75 (91–116) 2007  Geometric constructions of optimal linear perfect hash families Barwick, Susan; Jackson, WenAi, Finite Fields and Their Applications 14 (1–13) 2007  Projective aspects of the AES inversion Jackson, WenAi; Murphy, S, Designs Codes and Cryptography 43 (167–179) 2007  Projective ovoids and generalized quadrangles Brown, Matthew, Advances in Geometry 7 (65–81) 2007  Flock generalized quadrangles and tetradic sets of elliptic quadrics of PG(3, q) Barwick, Susan; Brown, Matthew; Penttila, T, Journal of Combinatorial Theory Series A 113 (273–290) 2006  Flux compactifications on projective spaces and the Sduality puzzle Bouwknegt, Pier; Evslin, J; Jurco, B; Varghese, Mathai; Sati, Hicham, Advances in Theoretical and Mathematical Physics 10 (345–394) 2006  Fractional analytic index Varghese, Mathai; Melrose, R; Singer, I, Journal of Differential Geometry 74 (265–292) 2006  On a generalised ConnesHochschildKostantRosenberg theorem Varghese, Mathai; Stevenson, Daniel, Advances in Mathematics 200 (303–335) 2006  The elliptic curves in gauge theory, string theory, and cohomology Sati, Hicham, The Journal of High Energy Physics (Print Edition) 3 (0–19) 2006  The polynomial degree of the Grassmannian G(1, n, q) of lines in finite projective space PG(n, q) Glynn, David; Maks, J; Casse, Rey, Designs Codes and Cryptography 40 (335–341) 2006  Arithmetic properties of eigenvalues of generalized harper operators on graphs Dodziuk, Josef; Varghese, Mathai; Yates, Stuart, Communications in Mathematical Physics 262 (269–297) 2005  Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in Ktheory Kordyukov, Y; Varghese, Mathai; Shubin, M, Journal fur die Reine und Angewandte Mathematik 581 (193–236) 2005  The index of projective families of elliptic operators Varghese, Mathai; Melrose, R; Singer, I, Geometry & Topology Online 9 (341–373) 2005  Dixmier traces as singular symmetric functionals and applications to measurable operators Lord, Steven; Sedaev, A; Sukochev, F, Journal of Functional Analysis 224 (72–106) 2005  A fundamental solution for linear secondorder elliptic systems with variable coefficients Clements, David, Journal of Engineering Mathematics 49 (209–216) 2004  Gerbes, Clifford Modules and the index theorem Murray, Michael; Singer, Michael, Annals of Global Analysis and Geometry 26 (355–367) 2004  Mtheory, type IIA superstrings, and elliptic cohomology Kriz, I; Sati, Hicham, Advances in Theoretical and Mathematical Physics 8 (345–394) 2004  Memory, market stability and the nonlinear cobweb theorem Gaffney, Janice; Pearce, Charles, The ANZIAM Journal 45 (547–555) 2004  Optimal linear perfect hash families with small parameters Barwick, Susan; Jackson, WenAi; Quinn, Catherine, Journal of Combinatorial Designs 12 (311–324) 2004  Geometric means, index mappings and entropy Comanescu, D; Dragomir, S; Pearce, Charles, chapter in Inequality theory and applications  Volume 3 (Nova Science Publishers) 85–96, 2003  Geometric means, index mappings and supermultiplicativity Pearce, Charles; Dragomir, S; Comanescu, D, chapter in Inequality theory and applications  Volume 2 (Nova Science Publishers) 193–201, 2003  A boundary element method for the numerical solution of a class of elliptic boundary value problems for anisotropic inhomogeneous media Azis, Mohammad; Clements, David; Budhi, W, The ANZIAM Journal 44 (C79–C95) 2003  A dualreciprocity boundary element method for a class of elliptic boundary value problems for nonhomogenous anisotropic media Ang, W; Clements, David; Vahdati, N, Engineering Analysis With Boundary Elements 27 (49–55) 2003  Method of best successive approximations for nonlinear operators Torokhti, Anatoli; Howlett, P; Pearce, Charles, Journal of Computational Analysis and Applications 5 (299–312) 2003  On the ClarkOcone theorem for fractional Brownian motions with Hurst parameter bigger than a half Bender, C; Elliott, Robert, Stochastics and Stochastic Reports 75 (391–405) 2003  Approximating Spectral invariants of Harper operators on graphs II Varghese, Mathai; Schick, T; Yates, S, Proceedings of the American Mathematical Society 131 (1917–1923) 2003  Approximating spectral invariants of Harper operators on graphs Varghese, Mathai; Yates, Stuart, Journal of Functional Analysis 188 (111–136) 2002  Families index theory for Overlap lattice Dirac operator. I Adams, Damian, Nuclear Physics B 624 (469–484) 2002  Families index theory, gauge fixing, and topology of the space of latticegauge fields: a summary Adams, Damian, Nuclear Physics BProceedings Supplements 109A (77–80) 2002  Semiclassical asymptotics and gaps in the spectra of magnetic Schrdinger operators Varghese, Mathai; Shubin, M, Geometriae Dedicata 91 (155–173) 2002  The BorelWeil theorem for complex projective space Eastwood, Michael; Sawon, J, chapter in Invitations to geometry and topology (Oxford University Press) 126–145, 2002  Best approximation of operators in the modeling of nonlinear systems Torokhti, Anatoli; Howlett, P, IEEE Transactions on Circuits and Systems I  regular papers 49 (1792–1798) 2002  Direct computation of the performance index for an optimally controlled active suspension with preview applied to a halfcar model Thompson, A; Pearce, Charles, Vehicle System Dynamics 35 (121–137) 2001  On bestapproximation problems for nonlinear operators Howlett, P; Pearce, Charles; Torokhti, Anatoli, Nonlinear Functional Analysis and Applications 6 (351–368) 2001  Performance index for a preview active suspension applied to a quartercar model Thompson, A; Pearce, Charles, Vehicle System Dynamics 35 (55–66) 2001  Twisted index theory on good orbifolds, II: Fractional quantum numbers Marcolli, M; Varghese, Mathai, Communications in Mathematical Physics 217 (55–87) 2001  Conformally invariant differential operators on spin bundles Eastwood, Michael, chapter in Further advances in twistor theory. Vol. III, Curved twistor spaces (Chapman & Hall/CRC) 72–74, 2001  Martingale methods in dynamic portfolio allocation with distortion operators Hamada, M; Sherris, M; Van Der Hoek, John, Quantitative Methods in Finance (2001), Sydney, Australia 12/12/01  Dirac operator index and topology of lattice gauge fields Adams, David, Chinese Journal of Physics 38 (633–646) 2000  msystems of polar spaces and maximal arcs in projective planes Hamilton, N; Quinn, Catherine, Bulletin of the Belgian Mathematical SocietySimon Stevin 7 (237–248) 2000  More on the pizza theorem Pearce, Charles, Australian Mathematical Society Gazette 27 (4–5) 2000 
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