The Department of Pure Mathematics seminar schedule is below. Unless stated otherwise, seminars are held in the Red Centre room RC-4082.

For further information or to be included on the email announcement list for this seminar, please contact Brian Jefferies.

Title: Packing spanning trees in graphs and bases in matroids.
Speaker: Dr. Robert Bailey (Carleton University)
Abstract:
The spanning tree packing number of a graph G, denoted σ(G), is the largest number of edge-disjoint spanning trees in G. An obvious upper bound on σ(G) is the edge-connectivity of G. We consider those graphs for which these two parameters are equal, and obtain a constructive description of them. We can also ask an equivalent question for matroids, and will conclude by mentioning this.

This is joint work with Brett Stevens (Carleton) and Mike Newman (Ottawa).

Title: Some versions of Khintchine inequality in rearrangement invariant spaces.
Speaker: Prof. Sergey Astashkin (Samara State University)
Abstract:
The main result we will discuss is the following: the square function inequality holds in an rearrangement invariant space X for all sequences of independent mean zero random variables from X if and only if X has the Kruglov property. The same condition is necessary and sufficient for a version of the well-known Maurey's inequality for vector-valued Rademacher series with independent coefficients to hold in X.

Title: Modifying Möbius.
Speaker: Mr. Peter Brown (UNSW)
Abstract:

Title: The behavior of functions of operators under perturbations.
Speaker: Prof. Vladimir Peller (Michigan State University)
Abstract:
I am going to speak about recent joint results with A. B. Aleksandrov. It is well known that a Lipschitz function does not have to be operator Lipschitz. In other words, the inequality |f(x)-f(y)| ≤ const |x-y| does not imply that ||f(A)-f(B)|| ≤ const ||A-B|| for self-adjoint operators A and B. It turned out that the situation dramatically changes if we consider functions in Hoelder--Zygmund classes. We prove that if 0 ≤ α ≤ 1 and f is in the Hoelder class Λ_α(R), then for arbitrary self-adjoint operators A and B with bounded A-B, the operator f(A)-f(B) is bounded and ||f(A)- f(B)|| ≤ const ||A-B||^α. We prove a similar result for functions f of the Zygmund class Λ₁(R): ||f(A+K)-2f(A)+f(A-K)|| ≤ const||K||, where A and K are self-adjoint operators. Similar results also hold for all Hoelder-Zygmund classes Λ_α(R), α > 0. We also study properties of the operators f(A)-f(B) for f in Λ_α(R) and self-adjoint operators A and B such that A-B belongs to the Schatten--von Neumann class S_p. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.

Title: Combinatorics of partitions and the geometry of sheaves.
Speaker: Dr. Balazs Szendroi (University of Oxford)
Abstract:
This will be a whirlwind tour of some topics of current interest in higher dimensional geometry and its interface with some aspects of theoretical physics. I will start with elementary properties of the generating function of partitions, well known since Euler, and discuss how this series as well as various relatives arise in modern geometry via sheaf theory. There will be a passing mention of modularity, the electromagnetic duality of gauge theories, the classification of Platonic solids and infinite-dimensional Lie algebras; I will aim to demostrate how all these topics fit together via geometry and gauge theory.

Title: The Riemann zeta function, the Laplacian, and how to recover Lebesgue integration from the pole of a zeta function (Part 1).
Speaker: Dr. Steven Lord (Adelaide)
Abstract:
The Riemann zeta function is most famous for the, as yet undtermined, zeros of its analytic continuation. Much less focus is given to the residue of the simple pole of the Riemann zeta function at z=1 and the relation to the harmonic series.

The first part, of this two part talk, introduces a generalised notion of the Riemann zeta function using compact operators on a separable Hilbert space. We show how the residue of the zeta function of a compact operator can be identified with the Dixmier trace, a non-normal trace used as the foundation for the "noncommutative integral" in Alain Connes' theory of Noncommutative Geometry. We highlight the contributions which Fedor Sukochev (UNSW) and his collaborators made to the area, and the role a joint paper between Fedor, myself and Aleksandr Sedaev (Vorenzh, Russia) has played in the categorisation of zeta functions.

The second part, given in the Analysis Seminar on Wednesday 8th April, considers specific zeta functions associated to the Laplacian of a compact Riemannian manifold. For example, the zeta function associated to the Laplacian on the circle is just a multiple of the Riemann Zeta Function. We introduce zeta functions weighted by bounded operators and show how, in recent work, we solved a problem concerning the "noncommutative integral" that has been open for 20 years.

Namely, we recover the Lebesgue integral of any bounded (and then any) integrable function as the residue of a zeta function. If time permits, we will introduce the integral on the "noncommutative torus" and show, using the same technique, that it can be recovered from the zeta functions associated to the "noncommutative Laplacian".

Title: Superpotential algebras.
Speaker: Dr. Balazs Szendroi (University of Oxford)
Abstract:
I will describe a construction of some non-commutative algebras of geometric significance arising in recent work in supersymmetric gauge theory, defined by relations on a free quiver algebra coming from a single cyclic element called the superpotential. These algebras often have pleasant homological properties, and have connections to toric geometry and the 3-dimensional McKay correspondence (among many other things).

Title: Universal Deformation Formula for Non-Abelian Group Actions.
Speaker: Dr. Victor Gayral (University of Reims)
Abstract:
In this talk, I will explain how to generalize Rieffel's deformation formula for a class of symplectic exponential solvable Lie group actions. This is based on quantization technics for rank one symplectic hermitian symmetric spaces of non-compact type, in the case where the associated homogeneous space is of group-type. After a short explanation of the geometric setup (that forces us to consider such groups only), I will explain how wavelette-analysis and provides powerful tools to establish suitable estimates in this situation. In particular, an interesting generalization of the Calderon-Vaillancourt theorem will be proven.

Title: Hyperbolic geometry is projective relativistic geometry.
Speaker: A./Prof. Norman Wildberger (UNSW)
Abstract:
Einstein's theory of relativity, as interpreted by Minkowski, introduced a new type of geometry based on the quadratic form x²+y²+z²-t². Surprisingly, in the hundred years since, pure geometry based on this quadratic form has seen relatively little development---we seem reluctant to leave the familiar Euclidean setting, despite encouragement from our physicist friends.

In this talk we will show how relativistic geometry allows a fresh development of hyperbolic geometry, using the projective view of Cayley, Beltrami and Klein, and the theory of Rational Trigonometry, suitably modified.

This results in a simpler and more elegant hyperbolic trigonometry, increased accuracy for computations, and many new theorems. Additionally the connections with spherical/elliptic geometry become more natural, and the theory extends to general fields.

Title: Singular reduction and quantization for lattice gauge models.
Speaker: Prof. Gerd Rudolph (Leipzig)
Abstract:
After a short introduction to the subject I will discuss the following points:

• The lattice gauge model as a Hamiltonian system with symmetries


• Singular Marsden Weinstein reduction of this system


• Canonical quantization (including remarks on the observable algebra and its representations)


• How to include nongeneric gauge orbit strata on quantum level?


• Open problems

Title: Some combinatorics of affine sl(n) crystals.
Speaker: Dr. Peter Tingley (Melbourne)
Abstract:
The crystals of the irreducible highest weight representations for affine $sl(n)$ have been studied extensively, and have various combinatorial models. We will discuss a few of these models, their structure, and how they are related. In particular, we will discuss a recent model where the vertices are indexed by cylindric plane partitions. In some cases this model leads to particularly clean statements.

Title: Maximal orders on algebraic surfaces.
Speaker: Prof. Rajesh Kulkarni (Michigan State University)
Abstract:
Maximal orders over discrete valuation rings were classically studied by many authors. In the more recent times, Michael Artin started the local study of maximal orders over two dimensional rings. Following his vision, Daniel Chan, Colin Ingalls and I have embarked on the study of maximal orders on projective surfaces. The classification of these orders up to the so called ramification data is now complete. We will review these developments of the last decade. Most of the talk will be a leisurely walk through basic ideas in the subject.

Title: Mathematics of Privacy.
Speaker: A./Prof. Ljiljana Brankovic (Newcastle)
Abstract:
Data mining is now becoming a matchless tool for research and strategic planning and are used by companies, governments and research institutions alike. It depends on massive databases often containing personal information but it is commonly assumed that only aggregate values and patterns will be made available to users and that no confidential individual values could be disclosed. There are two general approaches to ensure this: adding noise to the original data and restricting queries that can be asked of the database. In either case it may still be possible to "compromise" the database, that is, to compute individual values or other sensitive information from a suitable combination of aggregate values. In this talk we pay a special attention to the balance between the usability and the privacy of the individual records in the database and we focus on the interplay between mathematics and security. We show, for example, the connection between compromise-free query collections and graphs with least eigenvalue -2, and the relationship between maximal compromise-free query collection and a maximum antichain of a finite set.

Title: Spectral shift functions.
Speaker: Dr. Anna Skripka (Texas A&M University)
Abstract:
Spectral shift functions (SSF) provide information about a quantitative change of the spectrum of a self-adjoint operator under the influence of a self-adjoint perturbation. The first order SSF, called Krein's SSF, was introduced in [Lifshits '52 + Krein '53]. Since then, it has been well explored and found in various problems of mathematical physics; it can also be recognized as the scattering phase [Birman, Krein '62] and the spectral flow in a non-commutative geometry setting [Azamov, Carey, Sukochev '07]. The first order SSF can govern only the case of a trace class perturbation (or a trace class difference of the resolvents). In order to encompass more general perturbations, one needs to consider modified, higher order, spectral shift functions. The second order SSF is due to [Koplienko '84]; the spectral shift functions of order greater than two are under development. The talk will give a brief overview of the spectral shift functions, including recent results on the ones of order greater than two, obtained by the speaker in collaboration with K. Dykema.

Title: Diagonally cyclic latin squares.
Speaker: Dr. Nicholas Cavenagh (UNSW)
Abstract:
In this talk we mention some unexepected combinatorial equivalences of diagonally cyclic latin squares. We also outline a proof that any set of three cyclically generated transversals can be completed to a latin square of prime order p>7. This talk will be pitched to a generalist pure mathematics audience.

This is joint work with Adrian Nelson (USyd) and Carlo Hamalainen (UQ).

Title: Distance geometry in quasihypermetric spaces.
Speaker: Dr. Reinhard Wolf (Salzburg)
Abstract:
Let (X,d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X. Define I:M(X) --> R by I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), and set M(X) = sup I(\mu), where \mu ranges over the collection of signed measures in M(X) of total mass 1. We investigate the geometric constant M(X) and its relationship to metric properties of $X$, especially to the so-called quasihypermetric property.

Title: Stern Brocot trees, Ford circles, and Pell's equation.
Speaker: Assoc. Prof. Norman Wildberger (UNSW)
Abstract:
One of the most striking elementary results of twentieth century mathematics was Lester Ford's discovery of remarkable circles associated to fractions. They are closely related to the Stern Brocot tree, named after a nineteenth century German mathematician and French clockmaker, and also to Farey sequences. We'll see how they also connect to a highly anorexic tiling of the first quadrant into `wedges', and (briefly) also to hyperbolic geometry.

Then we'll explain how the Stern Brocot tree allows us a simpler way to solve that most important of Diophantine equations: Pell's equation. This talk will be elementary, and suitable for undergraduates.

Title: Cohomology of infinitesimal algebraic groups and quantized enveloping algebras.
Speaker: Christopher Drupieski (Virginia)
Abstract:
In 1984, Andersen and Jantzen computed the structure of the cohomology ring with trivial coefficients of the restricted Lie algebra corresponding to a reductive algebraic group. In 1993, Ginzburg and Kumar adapted the arguments of Andersen and Jantzen to compute the cohomology ring with trivial coefficients of the finite-dimensional quantum enveloping algebra at an ell-th root of unity defined by Lusztig. In both cases, lower bounds were assumed on the characteristic p of the ground field (respectively, the order ell of the root of unity) in order to achieve key vanishing results. Beyond some ad hoc calculations by Andersen and Jantzen, general results for small values of p and ell remained elusive.

In this talk I will report on some recent results of Bendel, Nakano, Parshall and Pillen that provide a uniform method for computing the aforementioned cohomology groups, as well as some recent results and some open questions in the quantum mixed case. Related results on the coordinate rings of nilpotent varieties may also be touched upon.

Some basic familiarity with group theory and Lie algebras will be assumed (e.g., root space decomposition). I will endeavor to present the remaining content on algebraic groups, Frobenius kernels, spectral sequences, and quantized enveloping algebras in a sufficiently elementary fashion that no previous mastery of these topics will be assumed.

Title: Why it is useful to study linear algebra combinatorially.
Speaker: Dr. Thomas Britz (UNSW)
Abstract:
Almost a century ago, the study of the Four Colour Problem drove mathematicians to study linear algebra in terms of combinatorial properties. Matrices and projective spaces were thus replaced by more abstract objects, known as matroids, and it turned out that these matroids not only generalised parts of linear algebra but also large parts of graph theory, matching theory, and many other areas.

In this talk, I will present several ways in which classical results from coding theory may be generalized with respect to matroids. To illustrate why these generalizations are interesting, I will show applications to designs, graphs, some big polynomials, and an interesting dual identity.

The talk is intended for a general audience.

Title: Hyperbolic geometry: past and future.
Speaker: A. Prof Norman Wildberger (UNSW)
Abstract:
Non-Euclidean geometry was one of the great developments of nineteenth century mathematics. This talk will give a historical introduction to hyperbolic geometry, and then introduce a simpler and more elegant direction for the subject, based on extending the ideas of rational trigonometry and universal geometry.

We begin by mentioning the initial developments of Gauss, Bolyai and Lobachevski, the current models associated due to Beltrami and Poincare relying on the approach to geometry advocated by Riemann, and then moving to a somewhat forgotten chapter --- the projective model associated to Cayley, Beltrami and Klein. This is the jumping off point to our new approach, which we will only have time to briefly discuss.

This talk will be very low on formulas and high on pictures, and should be easily accessible to undergraduates.

Title: The Equivariant Brauer Group.
Speaker: Prof. Dana Williams (Dartmouth College)
Abstract:
In algebraic topology, we learn to associate groups Hⁿ(T) to locally compact spaces which "count the n-dimensional holes in T'". In this talk, I want to describe how to realize H³(T) as a set Br(T) of equivalence classes of certain well-behaved C*-algebras. The group structure imposed on Br(T) via its identification with H³(T) is very natural in its C*-setting. With this group structure, Br(T) is called the Brauer group of T. Depending on your point of view, this result can be viewed either as a concrete realization of H³(T) or as a classification result for a class of C*-algebras. In the last part of the talk, I want to describe an equivariant version of Br(T) developed jointly with David Crocker, Alex Kumjian and Iain Raeburn. No prior knowledge of C*-algebras or operator algebras will be assumed.

Title: The Toeplitz algebra of an ax+b semigroup.
Speaker: Prof. Iain Raeburn (Wollongong)
Abstract:
Cuntz has recently constructed a very interesting simple C*-algebra from the ax+b semigroup over the natural numbers. We will discuss joint work with Marcelo Laca in which we study the (much larger) Toeplitz algebra of this semigroup, which turns out to be quasi-lattice ordered in the sense of Nica.

We will briefly discuss quasi-lattice ordered groups, focusing on examples, and the general theory of their Toeplitz algebras. We run the ax+b example through this theory, and discover that Cuntz's algebra is an important distinguished quotient of the Toeplitz algebra.

If we have time we will discuss the KMS states for a canonical dual action of the real numbers.

Title: Noncommutative blowing up.
Speaker: Kenneth Chan (UNSW)
Abstract:
In algebraic geometry, blowing up is a procedure which produces a new variety Z' from a variety Z by magnifying infinitesimal data. Noncommutative generalisations of blowing up arose in Chan and Ingall's program for resolving noncommutative singularities and in the work of M. van den Bergh in a somewhat different context. We will show via some simple examples that these noncommutative incarnations of blowing up coincide.

Title: Trace Shifts and C*-algebras.
Speaker: Dr. Teresa Bates (UNSW)
Abstract:
In the 1970's Mazurkiewicz introduced trace languages as a model for parallel processing in theoretical computer science. In this talk we discuss shift spaces associated to these languages which we call trace shifts. We will begin by recalling some basic concepts from the theory of symbolic dynamics, and move on to discuss the construction of trace languages and trace shifts. We will give an algorithm for constructing trace shifts out of directed graphs. To finish we will briefly describe the C*-algebras associated with trace shifts and some of their properties. This talk assumes no prior knowledge of symbolic dynamics, and the C*-algebra content will be minimal.

This is joint work with David Pask of the University of Wollongong.

Title: The D_2n planar algebras.
Speaker: Dr. Scott Morrison (Microsoft Station Q)
Abstract:
I'll give a gentle introduction to the world of "subfactor planar algebras". These are lovely topological/combinatorial gadgets, which describe the representation theory of subfactors. There's an "ADE" classification of the small examples, and I'll quickly focus on the type D cases. Here, I'll
show you how to give explicit generators and relations for the planar algebras. Perhaps at the end of the talk I'll have time to explain how to use these examples to define invariants of knots, and then to obtain strange new identities between classical knot polynomials.

Title: Symmetric norms and spaces of operators.
Speaker: Prof. Fedor A. Sukochev (UNSW)
Abstract:
In 1937, von Neumann showed that if ||.||_E is a symmetric norm on R^n then one can define a norm on the space of nxn matrices by

||A||_E = ||(s_1(A),...,s_n(A)||_E

where s_1(A),...,s_n(A) are the singular values of A (i.e. the eigenvalues of (A*A)^(1/2) in decreasing order. Surprisingly, the infinite-dimensional analogue of this result, although well-known in special cases, has never been established in complete generality. Very recently, in a joint work with N. Kalton, we have shown that if (E,||.||_E) is a symmetric Banach sequence space then the corresponding space S_E of operators on a separable Hilbert space, defined by T in S_E if and only if (s_n(T))^infty_{n=1} in E, is a Banach space under the norm ||T||_{S_E} = ||(s_n(T))^infty_{n=1}||_E thus providing complete infinite-dimensional extension of von Neumann's result. The proof that ||.||_{S_E} is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebras with a normal faithful semi-finite trace.

Title: Meta-Fibonacci Recursions: An Invitation to Hofstadter, Conolly and Much More.
Speaker: Prof. Steve Tanny (Toronto)
Abstract:
The study of integer sequences defined by meta-Fibonacci recursions (also called "self-referencing" or "nested" recursions) is a promising new field that has grown in the last thirty years from a brief mention in Hofstadter's book to dozens of papers by mathematicians around the world. In these recursions the arguments in the function defining the sequence depend upon earlier values of the sequence itself.

In this talk we provide some general background on these recursions. We then discuss some generalizations of the Conolly recursion, defined by C(n) = C(n-C(n-1)) + C(n-1-C(n-2)), with initial conditions C(1) = C(2) = 1. In particular, we examine variants of the Conolly recursion, together with their corresponding initial conditions, that generate sequences that are slowly growing: that is, the sequences begin with 1, are monotone non-decreasing and successive terms differ by 0 or 1. In certain cases, we are able to derive a combinatorial interpretation for these sequences: we show that they count the number of leaves in certain sub-trees of infinite trees, including binary trees, with special labeling schemes.

Title: Surfaces in your backyard.
Speaker: Prof. Ulf Persson (Chalmers University of Technology)
Abstract:
Curves are simple and created by God, surfaces a mess, created by the Devil, at least according to Enriques, the founder of the modern classification theory of surfaces, the zoo of which is going to be represented with a few illustrative examples, picked so to speak from your backyard.

Title: Asymptotic enumeration of correlation-immune boolean functions.
Speaker: Dr. Catherine Greenhill (UNSW)
Abstract:
J. Riordan, a famous combinatorialist, wrote in 1968:

Combinatorialists use recurrence, generating functions, and such transformations as the Vandermonde convolution; others, to my horror, use contour integrals, differential equations, and other resources of mathematical analysis.

An example of the kind of approach that would have horrified Riordan is the use of multidimensional complex integration to solve combinatorial counting problems. I will outline a recent example of this approach, involving the asymptotic enumeration of correlation-immune boolean functions. These functions are of great interest due to their cryptographic properties, and are related to orthogonal arrays used in statistics.

Joint work with Rod Canfield (Georgia), Jason Gao (Carleton), Brendan McKay (ANU) and Bob Robinson (Georgia).

Title: C^*-algebras associated to coverings of k-graphs.
Speaker: Prof. Alex Kumjian (University of Nevada)
Abstract:
A covering of k-graphs induces an embedding of universal C^*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realize a direct limit of $k$-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph (up to Morita equivalence). We will also discuss techniques for computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras.

Title: .
Speaker: Dr. Emine Sule Yaziki (Koc University, Turkey)
Abstract:
A t-(v,k, lambda) design D, for positive t, is an ordered pair (V,B), where V is a set of v elements, called points, B is a collection (multiset) of k-subsets of V, called blocks, such that each t-subset of V belongs to exactly lambda blocks. A defining set of a t-(v,k, lambda) design is a set of blocks which is a subset of a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper subsets is a defining set. A smallest defining set is one with smallest cardinality. This talk will summarise the ideas of earlier algorithms and proposes a new and more efficient algorithm that finds all non-isomorphic minimal defining sets of a given t-design. The complete list of minimal defining sets of the full 2-(7,3,5) design, 2-(15,3,1) designs, 2-(25,5,1) design and 2-(31,6,1) design were found. I will also give some theoretical results on the spectrum for full designs.

Title: Strichartz estimates.
Speaker: Robert Taggart (UNSW)
Abstract:
Strichartz estimates are spacetime estimates for solutions to wave-like equations, such as the accoustic wave equation and Schrodinger's equation. In this talk, we give an historical introduction to Strichartz estimates and illustrate how they are used to establish the well-possedness of semilinear PDEs. Finally, we present a new abstract theorem which extends results due to Terry Tao, Mark Keel and Damiano Foschi. This theorem implies new Strichartz estimates for the wave and Klein-Gordon equations.

Title: Gromov's dimension comparison theorem for Carnot groups.
Speaker: Dr. Ben Warhurst (UNSW)
Abstract:
A Carnot group G is naturally equipped with equivalent Euclidean and subriemannian metrics. Gromov's original question for submanifolds is as follows: Determine all possible pairs (α, β) in R² such that there exists a submanifold M a subset of G with Euclidean Hausdorff dimension α and Subriemannian Hausdorff dimension β. The difficulty in answering this question stems from the nonintegrability of the horizontal distribution in TG from which the subriemannian metric is derived. If we only require M be a subset of G then a complete answer to Gromov's question can be formulated. The solution uses elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. This is the result of joint work with Zoltan Balogh (Bern) and Jeremy Tyson (Illinois).

Title: Gregarious p-cycle decompositions.
Speaker: Benjamin R. Smith (The University of Queensland)
Abstract:
A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Necessary conditions for the existence of such a decomposition of complete equipartite graphs (having all parts the same size) are easily obtained. We discuss progress made toward establishing the sufficiency of these conditions in the case where the cycle length is a prime p.

Title: Decompositions of complete graphs with application to neighbourly translations of d-cubes.
Speaker: A./Prof. Elizabeth J Billington (The University of Queensland)
Abstract:
Currently there is no purely graphical proof of the fact that a complete graph K_n which has an edge-disjoint decomposition into t bipartite graphs must satisfy t ≥ n-1. (This is known as Graham and Pollack's theorem.)

I shall give one of the known proofs of this result, using simple linear algebra, and include related results by Hoffman and de Caen on possible types of bipartite graphs in the extreme case, with equality above.

An application to neighbourly d-cubes will also be given.

Title: Codes, matroids, designs, and graphs. And big polynomials.
Speaker: Dr. Thomas Britz (UNSW)
Abstract:
Matroids are combinatorial objects that generalise mathematical objects from linear algebra, graph theory, matching theory, and many other areas.

In this talk, I will present several ways in which classical results from coding theory may be generalized with respect to matroids, and I will explain why such generalizations are interesting. Designs, graphs, and some big polynomials will be making guest appearances.

The talk is intended for a general audience.

Title: Crossed products and k-graph algebras.
Speaker: Dr. Aidan Sims (University of Wollongong)
Abstract:
Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs. Higher-rank graphs and their C^*-algebras were developed by Kumjian and Pask in 2000 to model Robertson and Steger's higher-rank Cuntz-Krieger algebras. These higher-rank Cuntz-Krieger algebras are examples of Kirchberg algebas with torsion in K₁, and hence cannot be realised as graph algebras.

It has recently come to light that k-graph algebras can also be used to model simple, stably finite C^*-algebras which are not AF, and hence also not realisable as graph algebras. In fact, these examples are crossed products of AF graph algebras by the integers Z. In this talk we will outline the procedure for realising certain classes of crossed products of graph algebras by Z as k-graph algebras, and illustrate the ideas with some key examples.

Title: Discriminants of Braeur algebras
Speaker: Prof. Hebing Rui (East China Normal University)
Abstract:
Brauer algebras were introduced by Richard Brauer in order to study the tensor representations of the defining space for orthognal or symplectic groups. In 1995, Graham and Lehrer proved that Brauer algebras over a commutative ring are cellular algebras. A natural question is how to compute the Gram determinant for each cell module of a Brauer algebra. In this talk, I will anwser this question by giving a recursive formula. Such a formula has been generalized to Birman-Wenzl algebras, the q-analogue of Brauer algebras. We also give necessary and sufficient condition for the semisimplicity of Birman-Wenzl algebras over an arbitrary field which improves a work by H. Wenzl in 1990.

This is a joint work with Mei Si.

Title: Superintegrable systems and algebraic varieties.
Speaker: Dr. Jonathan Kress (UNSW)
Abstract:
Maximally superintegrable systems are Hamiltonian systems having the maximum number of continuous constants of the motion. This requirement forces bound trajectories to be closed and periodic and is a feature of the well known harmonic oscillator and Kepler-Coulomb systems. Other classes of maximally superintegrable systems are known, but most studies have been restricted to those that can be found by separation of variables.

This talk will discuss a method for finding and classifying superintegrable systems that does not rely on separation of variables. Each system in two-dimensional complex Euclidean space, having constants quadratic in the momenta, is found to correspond to a point on a 3-dimensional algebraic variety in 6 complex variables. The action of the Euclidean group is then used to distinguish inequivalent systems.

Title: Chromogeometry---a three fold symmetry in planar geometry.
Speaker: A./Prof. Norman Wildberger (UNSW)
Abstract:
Together with familiar Euclidean geometry, there are two relativistic planar geometries that form something of a holy trinity. Numerous theorems of classical geometry have relativistic analogs, and by looking at all three geometries at once we get many surprising new insights.

This talk will show how the nine-point circle of a triangle may be augmented by two `nine-point hyperbolas', how to find the `diagonals' and `corners' of an ellipse and why in general we should expect an ellipse to have TWO pairs of foci and directrices, why the parabola is the most distinguished of all the conics, and how various centers of quadrilaterals interact to give area invariance theorems.

The theory holds over a general field, in the spirit of rational trigonometry.

Title: Finite dimensional algebras and quantum groups.
Speaker: A./Prof. Jie Du (UNSW)
Abstract:
I am going to present an overview of the book with the given title. This research/advanced graduate level text combines for the first time in book form the two theories in the title, and its authors form a large Pacific triangle (Sydney-ShanghaiBeijing-Charlottesville-Sydney).

I will start with (generalized) Cartan matrices and their graph and root datum realizations. With the graph realization, I will discuss briefly representations of directed graphs, Gabriel's Theorem and Ringel-Hall algebras. With the root datum realization, I will introduce the associated quantum groups which are regarded as the latest development in Lie theory. I will conclude with the two exciting realizations: The Ringel-Hall algebra realization of the ±-part, and the Beilinson-Lusztig-MacPherson realization of the entire quantum gl_n via quantum Schur algebras.

Title: An uncertainty principle for operators.
Speaker: Prof. Michael Cowling (UNSW)
Abstract:
There are various uncertainty principles for functions, including Hardy's: if f is a function on R, such that |f(x)| ≤ e^-α|x|2 and |\hat f(ξ)| ≤ e^{-β |ξ|2}, where αβ > 1/4, then f = 0.

We prove a similar principle for kernel operators.

Title: Projection constants for products of families of projections.
Speaker: Assoc. Prof. Ian Doust (UNSW)
Abstract:
Two commuting families of projections {P_j} and {Q_k} on a Banach space X generate a finer decomposition. In many problems in functional analysis, and in particular in spectral theory, it is important to know how the projection constants for the finer decomposition depend on those of the original families {P_j} and {Q_k}. In this talk I will survey what is known in this area, highlighting some of the very recent progress that has been made.

Title: Neuberg cubics, elliptic curves and Desmic structure over finite fields.
Speaker: A./Prof. Norman Wildberger (UNSW)
Abstract:
Elliptic curves are cubics with a group structure which may be defined geometrically. These are naturally connected with many questions in number theory, topology and analysis, and have been extensively investigated. The Neuberg cubic of modern triangle geometry is arguably the most comprehensive organizational structure for a triangle, connecting dozens---even hundreds---of triangle centers and associated lines.

In this talk we gently introduce you to the above ideas, bring in the Desmic or linking structure so named by John Conway, and show how triangle geometry might shed light on aspects of elliptic curves over finite fields.

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Speaker: Prof. Estate Khmaladze (Victoria University of Wellington)
Abstract: