Past

We report on work joint with Scott Parsell in which estimates are obtained for the set of real numbers not closely approximated by a given form with real coefficients. "Slim"

technology plays a role in obtaining the sharpest estimates.

This is joint work with Diaz, Glesser and Park.

In Proc. Instructional Conf, Oxford 1969, G. Glauberman shows that

several global properties of a finite group are determined by the properties

of its p-local subgroups for some prime p. With Diaz, Glesser and Park, we

reviewed these results by replacing the group by a saturated fusion system

and proved that the ad hoc statements hold. In this talk, we will present

the adapted versions of some of Glauberman and Thompson theorems.

Multicore chips are nearly ubiquitous in modern machines, and to fully exploit this continuation of Moore's Law, numerical algorithms need to be able to exploit parallelism. We describe recent approaches to both dense and sparse parallel Cholesky factorization on shared memory multicore systems and present results from our new codes for problems arising from large real-world applications. In particular we describe our experiences using directed acyclic graph based scheduling in the dense case and retrofitting parallelism to a

sparse serial solver.

We introduce the notion of $G$-Higgs bundle from studying the representations of the fundamental group of a closed connected oriented surface $X$ in a Lie group $G$. If $G$ turns to be the isometry group of a Hermitian symmetric space, much more can be said about the moduli space of $G$-Higgs bundles, but this also implies dealing with exceptional cases. We will try to face all these subjects intuitively and historically, when possible!

In this talk, I shall endeavour to explain to the uneducated and uninitiated the joys and pleasures one can have studying automata.

Bridgeland's notion of stability condition allows us to associate a complex manifold, the space of stability conditions, to a triangulated category $D$. Each stability condition has a heart - an abelian subcategory of $D$ - and we can decompose the space of stability conditions into subsets where the heart is fixed. I will explain how (under some quite strong assumpions on $D$) the tilting theory of $D$ governs the geometry and combinatorics of the way in which these subsets fit together. The results will be illustrated by two simple examples: coherent sheaves on the projective line and constructible sheaves on the projective line stratified by a point and its complement.

Parity games are simple two-player, infinite-move games particularly useful in Computer Science for modelling non-terminating reactive systems and recursive processes.  A longstanding open problem related to these games is whether the winner of a parity game can be decided in polynomial time.  One of the most promising algorithms to date is a strategy improvement algorithm of Voege and Jurdzinski, however no good bounds are known on its running time.

In this talk I will discuss how the problem of finding a winner in a parity game can be reduced to the problem of locally finding a global sink on an acyclic unique sink oriented hypercube.  As a consequence, we can improve (albeit only marginally) the bounds of the strategy improvement algorithm.

This talk is similar to one I presented at the InfoSys seminar in the Computing Laboratory in October.

This talk will give detailed proofs of Szemer\'{e}di's Regularity Lemma for graphs and the deduction of Roth's Theorem. One can derive Szemer\'{e}di's Theorem on arithmetic progressions of length $k$ from a suitable regularity result on $(k-1)$-uniform hypergraphs, and this will be introduced, although not in detail.

It is well known that the description of the asymptotic behaviour of products of i.i.d random matrices can be derived from the properties of the Lyapunov exponents of these matrices. So far, the fact that the matrices in question are IDENTICALLY distributed, had been crucial for the existing theories. The goal of this work is to explain how and under what conditions one might be able to control products of NON-IDENTICALLY distributed matrices.

We present a new class of self interacting Markov chain models. In contrast to traditional Markov chains, their time evolution may depend on the occupation measure of the past values. We propose a theoretical basis based on measure valued processes and semigroup technics to analyze their asymptotic behaviour as the time parameter tends to infinity. We exhibit different types of decays to equilibrium depending on the level of interaction. In the end of the talk, we shall present a self interacting methodology to sample from a sequence of target probability measures of increasing complexity. We also analyze their fluctuations around the limiting target measures.

he study of polycrystals of shape-memory alloys and rigid-perfectly plastic materials gives rise to problems of nonlinear homogenization involving degenerate energies. We present a characterisation of the strain and stress fields for some classes of problems in plane strain and also for some three-dimensional situations. Consequences for shape-memory alloys and rigid-perfectly plastic materials are discussed through model problems. In particular we explore connections to previous conjectures characterizing those shape-memory polycrystals with non-trivial recoverable strain.

Abstract: In this talk we show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case. We show how to associate a quantum state to the I-brane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, our formalism elegantly reconstructs the dual Nekrasov-Okounkov partition function from a quantum Seiberg-Witten curve.

The paper shows that financial market equilibria need not exist if agents possess cumulative prospect theory preferences with piecewise-power value functions. The reason is an infinite short-selling problem. But even when a short-sell constraint is added, non-existence can occur due to discontinuities in agents' demand functions. Existence of equilibria is established when short-sales constraints are imposed and there is also a continuum of agents in the market