The Department of Pure Mathematics seminar schedule is below. Unless stated otherwise, seminars are held in the Red Centre room RC-3084.

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

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: 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.

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