Past

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

Solving completely $x+y-z=1$ in unknowns taken from the group generated by a variable $t$ with $1-t$ over a finite field is not so easy as might be expected. We present a generalization to arbitrary linear varieties and finitely generated groups (keywords effective Mordell-Lang). We also mention applications to (a) solving equations like $u_n+v_m+w_l+f_k=0$ in $n,m,l,k$ for given recurrences $u,v,w,f$; and to (b) finding the smallest order of non-mixing of a given algebraic ${\bf Z}^s$-action. This is joint work with Harm Derksen.

As research in topology optimisation has reached a level of maturity, two main classes of methods have emerged and their applications to real engineering design in industry are increasing. It has therefore become important to identify the limitations and challenges in order to ensure that topology optimisation is appropriately employed during the design process whilst research may continue to offer a more reliable and fast design tool to engineers.

The seminar will begin by introducing the topology optimisation problem and the two popular finite element based approaches. A range of numerical methods used in the typical implementations will be outlined. This will form the basis for the discussion on the short-comings and challenges as an easy-to-use design tool for engineers, particularly in the context of reliably providing the consistent optimum solutions to given problems with minimum a priori information. Another industrial requirement is a fast solution time to easy-to-set-up problems. The seminar will present the recent efforts in addressing some of these issues and the remaining challenges for the future.

I will describe a topological approach to the motion planning problem of

robotics which leads to a new homotopy invariant of topological spaces

reflecting their "navigational complexity". Technically, this invariant is

defined as the genus (in the sense of A. Schwartz) of a specific fibration.

I consider the problem of maximising the final utility of a portfolio which is constrained to satisfy the draw-down condition, i.e. the current value of the portfolio can not drop below a pre-specified funciton of its running maximum. It turns out that martingale techniques yield an explicit and rather elegant solution. The so- called Azema-Yor processes appear naturally and I take some time to introduce this class and discuss some of their remarkable properties.

In particular, I show how they can be characterised as (unique,

strong) solutions to SDEs called the Bachelier Eq and the Draw-Down Eq.

The talk is based (in particular) on a joint work with L. Carraro, N.

El Karoui and A. Meziou.

In this talk I will discuss some elementary notions of deformation theory in algebraic geometry like Schlessinger's Criterion. I will describe obstructions and deformations of sheaves in detail and will point out relations to moduli spaces of sheaves.

Recurrences have been central to the study of dynamical

systems ever since the inception of the subject. Periodic solutions

make the notion of recurrences exact. The Lorenz attractor is the best

known example of a strange attractor and we will describe a method to

find periodic solutions that lie on it. Additionally, we will consider

a turbulent channel flow and describe the computation of time periodic

solutions using nearly $300,000$ degrees of freedom to represent the

velocity field.

The talk will survey old and recent applications of variational techniques in studying the existence, stability and bifurcations of time harmonic, localized in space solutions of the nonlinear Schroedinger equation (NLS). Such solutions are called solitons, when the equation is space invariant, and bound-states, when it is not. Due to the Hamiltonian structure of NLS, solitons/bound-states can be characterized as critical points of the energy functional restricted to sets of functions with fixed $L^2$ norm.

In general, the energy functional is not convex, nor is the set of functions with fixed $L^2$ norm closed under weak convergence. Hence the standard variational arguments fail to imply existence of global minimizers. In addition for ``critical" and ``supercritical" nonlinearities the restricted energy functional is not bounded from below. I will first review the techniques used to overcome these drawbacks.

Then I will discuss recent results in which the characterizations of bound-states as critical points (not necessarily global minima) of the restricted energy functional is used to show their orbital stability/instability with respect to the nonlinear dynamics and symmetry breaking phenomena as the $L^2$ norm of the bound-state is varied.