The Schools of Mathematics and Statistics at UNSW and Sydney University hold a Joint Colloquium. The schedule of talks is below.

For further information or to be included on the email announcement list for this seminar, please contact Brian Jefferies at UNSW or Emma Carberry at Sydney University.

Seminars at UNSW are held in the Red-Centre room RC-4082 and those held at Sydney University are held in the Carslaw building in the room indicated in the table below. Unless indicated, the Joint Colloquia are on Friday at 2--3pm.

Title: Variational Methods in Materials and Imaging.
Speaker: Prof. Irene Fonseca (Carnegie Mellon)
Abstract:
Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations.

In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics, thin films, and quantum dots will be addressed.

Title: aps of groups that send cosets to cosets.
Speaker: Prof. Michael Cowling (Birmingham)
Abstract:
In many cases, maps of groups to groups that send cosets (of subgroups) to cosets have strong algebraic properties. In other cases, they do not. This talk surveys what we know on this topic, and why it is of interest.

Title: Holomorphic coisotropic reduction.
Speaker: Prof. Justin Sawon (Colorado State University)
Abstract:
Coisotropic reduction can be regarded as a generalization of symplectic reduction. Given a symplectic manifold X of dimension 2n with symplectic form $\omega$, a submanifold Y of dimension at least n is coisotropic if $\omega|_Y$ has the smallest possible rank at every point of Y. The null directions of $\omega|_Y$ then induce a foliation F on Y and the space of leaves Y/F is a symplectic manifold of lower dimension.

In this talk we will consider coisotropic reduction in holomorphic symplectic geometry. The main difficulty is ensuring that the leaves of the foliation are compact, so that Y/F is well-defined. We will describe some examples and applications of holomorphic coisotropic reduction.

Title: Curvature flows and complex geometry.
Speaker: Prof. Gang Tian (Princeton)
Abstract:
In this general talk, I will discuss how curvature flows can be applied to studying complex geometric problems. I will show interactions between singularity formation of curvature flows and classification of algebraic manifolds. I will also discuss some open analytic problems which arise from geometric applications.

Title: Weighted norm inequalities for pseudo-differential.
Speaker: Dr. David Rule (University of Edinburgh)
Abstract:
I'll give a brief introduction to pseudo-differential operators and their usefulness in partial differential equations. This will then lead on to some recent work with Nick Michalowski and Wolfgang Staubach concerning operators with symbols which are very rough in the spatial variable.

Title: Singular Integral Operators and Boundary Value Problems: Recent Progress.
Speaker: Prof. Marius Mitrea (University of Missouri)
Abstract:
I will review some of the historical developments in the area at the interface between Partial Differential Equations and Harmonic Analysis, with emphasis on the role played by boundary integral methods in the treatment of elliptic problems. The goal is to highlight the significance of tools from Harmonic Analysis which have played a fundamental role in establishing progressively sharper results in this field. The lecture is largely self-contained and
accessible to nonspecialists.

Title: How to choose random cubes, and why?
Speaker: Prof. Tuomas Hytönen (University of Helsinki)
Abstract:
A standard tool in mathematical analysis is the refining sequence of partitions of the Euclidean space into cubes with side-lengths equal to powers of two, the so-called dyadic cubes. Associated to the system of dyadic cubes, there is a natural basis of square-integrable functions, the Haar system, which has proven useful e.g. in the analysis of singular integral operators.

In their work on singular integrals related to quite general measures, Nazarov, Treil and Volberg observed that, while certain matrix coefficients of such an operator with respect to the Haar basis do not admit any good control, such bad situations only occur "rarely", which can be made precise in the sense of probability by working with a randomly chosen dyadic system, instead of a fixed one. I will discuss how these random cubes a chosen in the original Euclidean framework, and also what they could be in a more general metric space.

Title: Complexity of 3-manifolds.
Speaker: Prof. Stephan Tillmann (University of Queensland)
Abstract:
The complexity of a 3-manifold is the minimum number of tetrahedra in a triangulation of the manifold. It was defined and first studied by Matveev in 1990. The complexity is generally difficult to compute, and various upper and lower bounds have been derived during the last decade using fundamental group, homology or hyperbolic volume.

Effective bounds have only been found recently in joint work with Jaco and Rubinstein. Our bounds not only allowed us to determine the first infinite classes of minimal triangulations of closed 3-
manifolds, but they also lead to a structure theory of minimal triangulations.

In this talk, I will give a general introduction to complexity of 3-manifolds and explain its ramifications for algorithmic problems in the study of 3-manifolds.

Title: Homogeneous right inverses and metric projections.
Speaker: Dr Sergey Ajiev (UNSW)
Abstract:
Some problems, for example, in PDE can be reduced to the invertibility of a closed operator with a non-trivial and non-complemented kernel defined, for example, on a Sobolev space. We construct homogeneous (non-linear) right-inverse operators and establish explicit estimates for the exponents of their Hölder regularity and the corresponding Hölder seminorms in both abstract and particular settings. In the setting of Banach spaces, our right inverses provide optimal solution for the equation Af=g.

One deals with various types of Besov and Lizorkin-Triebel spaces of functions on an arbitrary open subset of an Euclidean space defined in terms of local approximations, differences, wavelet expansions, or a functional calculus and also with their duals, subspaces, quotients and a wide class of "independently generated spaces".

We also discuss the relations with the classical problem of the regularity of the metric projection map and establish the global regularity of this map in a quantitative manner. Attention is paid to the sharpness of some results, natural generalisations and the limitations of the linear and Lipschitz settings.

Title: The Kato square root problem - a new approach with applications to boundary value problems.
Speaker: Prof Alan McIntosh (CMA, ANU)
Abstract:
About 1960 Tosio Kato, during his investigation of the evolution of physical systems, was led to pose a key question about the square root of non-symmetric divergence form elliptic partial differential operators. The one-dimensional problem was solved by Coifman, Meyer and myself in 1982, while it was only in 2001 that the question was fully answered by Auscher, Hofmann, Lacey, Tchamitchian and myself. A more general viewpoint was subsequently provided in joint work of mine with Axelsson and Keith.

Title: Stochastic Landau-Lifschitz-Gilbert Equation.
Speaker: Prof. Zdzislaw Brzezniak (York, UK)
Abstract:
In this talk we will consider the three-dimensional Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise. This equation, fundamental for the theory of magnetic memories, describes evolution of spins in ferromagnetic materials under the influence of thermal noise. The necessity to include the thermal noise in the equation was observed by Physicists in the early fifties but the rigorous mathematical theory was missing. Since the physical theory prescribes the fixed length of the spin vectors, any solution must take values in the sphere. This fact leads to interesting connections of this equation with the equation for the flow of harmonic maps. We show the existence of weak martingale solutions taking values in a sphere and discuss regularity of solutions. We will sketch proof of the existence of solutions and formulate some open problems. This is a joint work with Ben Goldys and Terry Jegaraj (both UNSW Sydney).

Title: Classical and Quantum Invariant Theory.
Speaker: Prof. Gus Lehrer (Sydney)
Abstract:
Classical invariant theory, which has applications in diverse areas, such as Atiyah-Singer index theory, has been studied in 3 contexts, which are related in various ways. Despite its long history, there are still elementary unsolved problems. I shall
discuss these, and indicate some generalisations to (non-commutative) quantum invariants. (Joint work with R. Zhang and H. Zhang).

Title: Application of Homotopy Theory to Graph Colourings.
Speaker: Prof. Peter Zvengrowski (U. Calgary)
Abstract:
In 195 M. Kneser formulated a conjecture about the number of colours needed to colour certain graphs, which was proved in 1978 by L. Lovasz, using techniques of algebraic topology. In this talk we shall give the simplest version of the proof, which actually proves the more general Dol'nikov Theorem, and is due to J. Greene in 2002. The techniques used are fairly elementary, so all parts of the proof will be explained and should be accessible to students as well as faculty. If time permits some further research of the speaker related to these questions will be discussed.

Title: The "star-triangle" or "Yang-Baxter" relations in statistical mechanics.
Speaker: Prof. Rodney Baxter (ANU)
Abstract:
There are a number of lattice models, mostly two-dimensional, in statistical mechanics that have been solved exactly by using the star-triangle relation. By this I mean that the partition function per site and the spontaneous magnetization
have been calculated in the limit of a large lattice. I shall give a historical overview and discuss the remarkable invariance properties that follow from the star-triangle relation, indicating how they lead to the solutions. The talk is intended for a non-specialist audience.

Title: Knots and Algebraic Dynamical Systems.
Speaker: Prof. Susan G. Williams (University of South Alabama)
Abstract:
We study a classical invariant of knots, the Alexander module, via its Pontrjagin dual. This is an algebraic dynamical system, a compact group with an action by automorphisms. It has an elementary combinatorial description in terms of "dynamic" colourings of a knot diagram. We will discuss the relation between topology of the knot and dynamics of the dual, and give examples. (This is joint work with Daniel Silver.)

Title: Birational Geometry of Moduli Spaces of Curves.
Speaker: Prof. Ian Morrison (Fordham)
Abstract:
Moduli spaces are one of the beauties of algebraic geometry: sets of isomorphism classes of objects that turn out to carry a natural algebraic structure. Many general questions are of special interest for such moduli spaces and lead to a beautiful interplay between the geometry of the objects individually and in families. In my talk, I will try to introduce and illustrate these ideas. The moduli spaces I will discuss are those of algebraic curves---widely studied and applied in mathematical physics, symplectic geometry and number theory. The questions I will ask about them are from birational geometry and deal with maps from these spaces to complex projective spaces.

Title: Dynamical zeta functions and mixing.
Speaker: Prof. Mike field (Houston)
Abstract:
The dynamical zeta function is an analog of the Riemann zeta function. However, rather than being the Euler product over the prime numbers, the dynamical zeta function is a product over the prime periods of a flow. Just as happens in number theory, analytic and meromorphic properties of the dynamical zeta function encapsulate statistical properties of the flow such as the distribution of periodic orbits (prime number theorem) and rates of mixing. In this introductory talk we will describe some of the characteristic properties of dynamical zeta functions. We will also discuss the issue of exponential error estimates (which correspond to the Riemann hypothesis in number theory) as well as recent work on rates of mixing for hyperbolic flows including, we hope, new examples of smooth hyperbolic flows that stably mix exponentially fast.

Title: Fluid modelling of carbon dioxide sequestration.
Speaker: Prof. Herbert E Huppert (DAMTP, Cambridge)
Abstract:
Current global anthropogenic emissions of carbon dioxide are approximately 27 Gigatonnes annually. The influence of this green-house gas on climate has raised concern. A means of reducing environmental damage is to store carbon dioxide somewhere until well past the end of the fossil fuel era. Storage by injection of liquid, or supercritical, carbon dioxide into porous reservoir rocks, such as depleted oil and gas fields and regional saline aquifers, is being considered. The presentation will discuss the rate and form of propagation to be expected. It builds on theoretical and experimental investigations of input of liquid of one viscosity and density from a point source above an impermeable boundary, either horizontal or slanted, into a heterogeneous porous medium saturated with liquid of different viscosity and density. Key predictions are: 1) for constant supply the radius of carbon dioxide ponding below a horizontal impermeable barrier will increase as the square root of time; 2) at constant supply rate the central thickness of the carbon dioxide pond is invariant with time; 3) the radius is proportional to the quarter power of input flux and permeability; 4) the effect of a slope is unnoticed until a time scale which varies between months and years for typical natural parameters; and 5) it is possible to use measurements of radius to estimate volume stored. In the Sleipner natural gas field, carbon dioxide has been injected at a rate of ~ 1 Mt/yr since 1996. We will briefly show how to apply our results to interpret these field observations.

Title: What is the Thurston norm?
Speaker: Dr. Stephen Tillmann (Melbourne)
Abstract:
In the late seventies, Bill Thurston defined a semi-norm on the homology of a 3-dimensional manifold which lends itself to the study of manifolds which fibre over the circle. This led him to formulate the Virtual Fibration Conjecture, which is fairly inscrutable and implies almost all major results and conjectures in the field. Nevertheless, Thurston gave the conjecture "a definite chance for a positive answer" and much research is currently devoted to it.

Title: Noncommutative analysis and geometry.
Speaker: Prof. Fedor A Sukochev (UNSW)
Abstract:
Noncommutative analysis applies abstract methods of Banach space theory to spaces that naturally appear in operator theory. In the recent past, noncommutative analysis (in a wide sense) has developed rapidly because of its interesting and fruitful interactions with classical theories such as C* and W*-algebras, Banach spaces, probability, harmonic analysis. Important recent advance
in the differential calculus of functions of non-commuting operators has very recently led to new analytic formulae for the spectral flow along a smooth path of self-adjoint Breuer-Fredholm operators in a semi-finite von Neumann algebra. These analytic formulae for spectral flow of bounded/unbounded Breuer-Fredholm operators fit into the overall picture in noncommutative geometry and settle an issue which may be traced back to the 1974 Vancouver ICM address of I. M. Singer. This latter result is obtained jointly with A. Carey and D. Potapov.

Title: The irony of romantic mathematics.
Speaker: Assoc. Prof. Henrik Kragh Sørensen (University of Aarhus)
Abstract:
During the first part of the nineteenth century, mathematics underwent a number of important cognitive and institutional transformations. In this talk, I wish to illustrate and contextualise some of these transformations by contextualising a number of examples from the mathematical production of mathematicians such as N. H. Abel, C. F. Gauss, and N. Lobachevsky within the romantic period.

Many of the most famous and productive mathematicians of early nineteenth century were prototypical romantic heroes --- neglected geniuses who died young, suffering the material world while studying the immaterial mathematical entities. However, the romantic influence over mathematics during that period extended well beyond the purely biographical. Especially in the Germanic romantic era, mathematics was
immersed in a cultural embedding that will allow us to discuss perspectives on romantic irony from a mathematical viewpoint.

In the first part of the nineteenth century, mathematics developed in an increasingly conceptual direction. As part of this transition, mathematicians began asking fundamentally new kinds of questions that led to new types of answers. Instead of asking for explicit formulae as results, mathematicians began to question the very possibility of such formulae. At the same time, other discoveries (such as non-Euclidean geometry) led mathematicians to distance their pursuit from the investigation of nature, turning it into an autonomous discipline concerned with an immaterial mathematical realm.

Since the fifteenth century, mathematicians had searched for a general formula for solving equations of all degrees. However, around 1830 and coinciding with the late romantic period, the new concept-centred approach led innovative young mathematicians such as Abel and Galois to reformulate the question in terms of “solvability” rather than “solution”. Thereby, they shifted their focus to investigating the representability within certain (restricted) formal systems, yielding unforeseen results.

Title: Multi-normed spaces and multi-Banach algebras.
Speaker: Prof. Garth Dales (Leeds)
Abstract:
A standard first graduate course in functional analysis will cover Banach and Hilbert spaces, dual spaces, weak topologies, bounded linear operators on Banach spaces, and perhaps something on Banach lattices and on Banach algebras

I have developed a variant of this theory. In this one replaces the norm on a Banach space E by a sequence of norms, one on each of the spaces Eⁿ. This enables us to develop new results on the following topics, among others: (1) the geometry of Banach spaces and absolutely summing operators; (2) multi-continuous linear operators, generalizing the regular operators on a Banach lattice; (3) a more general theory of orthogonality in Banach spaces, and a new duality theory; (4) applications to modules over Banach algebras, especially L²(G) over the group algebra L¹(G) for a locally compact group G; (5) connections with the theory of amenable groups and algebras.

I will try to sketch some of these new theories.

Title: Variational approach for isoperimetric inequalities.
Speaker: Prof. Dorin Bucur (Université de Savoie)
Abstract:
In this talk, we are concerned with isoperimetic inequalities involving eigenvalues of elliptic operators. As an example, we discuss some classical problems, like the Faber-Krahn inequality for the first Dirichlet eigenvalue of the Laplacian, from both classical and variational points of view. We will point out the main ingredients for proving that the optimal shape is radial: prove the existence of an optimal domain, prove mild regularity of the free boundary, use a cut and
reflect argument in order to prove symmetry.

Recent advances and open problems will presented.

Title: Elliptic Curves and Multiple Zeta Numbers.
Speaker: Prof. Richard M Hain (Duke)
Abstract:
The multiple zeta value zeta(n_1,...,n_r) is defined by the convergent sum

zeta(n_1,...,n_r) = \sum_{0<k_1<...<k_r} 1/(k_1^{n_1}...k_r^{n_r})

where n_1,...,n_r are positive integers and n_r > 1. When the depth r is 1, they are simply the values of the Riemann zeta function at integers larger than 1. Depth 2 multiple zeta numbers were first considered by Euler and have recently resurfaced in the works of Zagier, Goncharov and others. Multiple zeta numbers occur as periods of the mixed Tate motives constructed by Deligne and Goncharov. They satisfy many interesting combinatorial identities; and some of their transcendence properties are controlled by the algebraic K-theory of the integers. After surveying these results, I will discuss some mysterious identities between depth 2 multiple zeta values that arise from cusp forms of SL_2(Z) that go back to Goncharov and are due to Gangl, Kaneko and Zagier. The final goal is tgive some idea of why elliptic modular forms should impose relations on multiple zeta numbers.

Title: Planar Algebra.
Speaker: Prof. Vaughen Jones (Berkeley)
Abstract:
I will talk about algebra having operations indexed by planar diagrams. This kind of algebra arises from various sources including knot theory, statistical mechanical models, matrix models, quantum groups, category theory and von Neumann algebras. There are many common themes including the Yang Baxter equation and other skein relations. A canonical construction of von Neumann algebras from planar algebra will be presented.

Title: Similar sublattices of the root lattice A₄.
Speaker: Prof. Michael Baake (Bielefeld)
Abstract:
(joint work with Manuela Heuer and Robert V. Moody)

Among the sublattices of a given lattice in Euclidean space, those similar to the original lattice form an interesting and important subclass. In recent years, several attempts have been made to classify them, with limited success so far. An exeption are lattices in dimensions up to 4, and some of the root lattices. The latter were studied by Conway, Rains and Sloane in a non-constructive manner by means of quadratic forms, which gave access to the possible sublattice indices, but not to the sublattices themselves.

In particular, the root lattice A₄ could not be treated completely. It is the purpose of this talk to add a constructive approach, based on the arithmetic of a certain quaternion algebra and the existence of an unusual involution of the second kind. This also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.

Title: Finiteness conditions and mapping tori
Speaker: Dr. J. A. Hillman (Sydney)
Abstract:
(joint work with D. H. Kochloukova, UNICAMP, Brazil)

The mapping torus of a self-homeomorphism f of a space X is the space obtained from the cylinder X x [0,1] by identifying the ends via f. (The Möbius band is a simple, nontrivial example of this construction.) There are good characterizations of n-manifolds which are mapping tori in all dimensions except n=4 or 5. We shall consider a homotopy analogue of the problem of recognizing mapping tori, in which "homeomorphism" and "manifold" are replaced by
"homotopy equivalence" and "Poincaré duality complex". Our argument is homological, and an essential element
is Kochloukova's result that certain Novikov extensions of group rings are weakly finite.

Our results are best possible, in a sense to be explained in the talk. In particular, we obtain the following 4-dimensional homotopy analogue of Stallings' characterization of 3-dimensional mapping tori:

if M' is an infinite cyclic covering space of a closed 4-manifold M then M' satisfies Poincaré duality with local coefficients if and only if χ(M)=0 and π₁(M')$ is finitely generated.

Title: A short history of Polish mathematics
Speaker: Prof. Wieslaw Zelazko (Warsaw)
Abstract:
Wieslaw Zelazko is Professor of Functional Analysis in the Institute of Mathematics of the Polish Academy of Sciences. He has a fine mathematical family tree including his academic father Stanislaw Mazur, grandfather Stefan Banach and great-grandfather Hugo Steinhaus. Professor Zelazko has received many honours and awards including the prestigious Stefan Banach Medal in 2000, and has served as President of the Polish Mathematical Society (1984-1986).

Title: Trees, Symmetric Spaces, Buildings and K-Theory for Group C*-Algebras.
Speaker: Prof. Paul Baum (Pennsylvania State University)
Abstract:
Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. Associated to the (left) regular representation of G is a C*-algebra known as the reduced
C*-algebra of G. In 1980 P. Baum and A. Connes conjectured an answer to the problem of calculating the K-theory of this C*-algebra. This conjecture -- when true -- has corollaries in various parts of mathematics and thus reveals connections between problems which previously appeared to be completely unrelated. This talk will state the conjecture and indicate its corollaries.

This talk is intended for non-specialists, so the basic definitions, i.e., C*-algebra, K-theory etc., will be carefully stated.