Past

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.

I will discuss the following

Conjecture B: Finitely generated abstract images of profinite groups are finite.

I will explain how it relates to the width of words and conjugacy classes in finite groups. I will indicate a proof in the special case of 'non-universal' profinite groups and propose several directions for future work.

This conjecture arose in my discussions with various participants of a workshop in Blaubeuren in May 2007 for which I am grateful. (You know who you are!)

In the first part of the talk we will introduce the notion of property testing and briefly discuss some results in testing graph properties in the framework of property testing.

Then, we will discuss a recent result about testing expansion in bounded degree graphs. We focus on the notion of vertex-expansion:   \newline an $a$-expander is a graph $G = (V,E)$ in which every subset $U$ of $V$ of at most $|V|/2$ vertices has a neighborhood of size at least $a|U|$. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time approximately $O(n^{1/2})$.

We design a property testing algorithm that accepts every $a$-expander with probability at least 2/3 and rejects every graph that is $\epsilon$-far from an $a^*$-expander with probability at least 2/3, where $a^* = O(a^2/(d^2 log(n/\epsilon)))$, $d$ is the maximum degree of the graphs, and a graph is called $\epsilon$-far from an $a^*$-expander if one has to modify (add or delete) at least $\epsilon d n$ of its edges to obtain an $a^*$-expander. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $O(d^2 n^{1/2} log(n/\epsilon)/(a^2 \epsilon^3))$.

This is a joint work with Christian Sohler.

In the Gulbenkian Lecture Theatre, St Cross Building, Manor Road.

Tea will be available in the Arumugam Building, St. Catherine's College, from 4.15pm.