Past

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

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.

We consider transient random walks in random environment on Z with zero asymptotic speed. In a seminal paper, Kesten, Kozlov and Spitzer proved that the hitting time of the level "n" converges in law, after a proper normalization, towards a positive stable law, but the question of the description of its parameter was left open since that time. A new approach to this problem, based on a precise description of Sinai's potential, leads to a complete characterization of this stable law, making a tight link with Kesten's renewal series. The case of Dirichlet environment turns out to be remarkably explicit. Quenched results on this model will be presented if time permits.

We present numerical schemes for nonlinear stochastic differential equations whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action and subsequently to the corresponding Lie algebra.

We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Lastly, we demonstrate our methods by presenting some numerical examples

We consider a class of energy functionals containing a small parameter ε and a long-range interaction. Such functionals arise from models for phase separation in diblock copolymers and from stationary solutions of FitzHugh–Nagumo type systems.

On an interval of arbitrary length, we show that every global minimizer is periodic, and provide asymptotic expansions for the periods.

In 2D, periodic hexagonal structures are observed in experiments in certain di-block

copolymer melts. Using the modular function and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.

We also consider the sharp interface problem arising in the singular limit,

and prove the existence and the nondegeneracy of solutions whose interface is a distorted circle in a two-dimensional bounded domain without any assumption on the symmetry of the domain.

Abstract: I will discuss some recent developments in understanding compactifications of the Heterotic string on Calabi-Yau manifolds. These compactifications are well-described by linear sigma models with (0,2) supersymmetry. I will show how to use these models to compute physical observables, such as genus zero Yukawa couplings, their singularity structure, and dependence on bundle moduli.

In this talk we present a work done with M. Di Giacinto (Università di Cassino - Italy) and Salvatore Federico (Scuola Normale - Pisa - Italy). The subject of the work is a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon.

The level of wealth is constrained to stay above a "solvency level".

The model is naturally formulated as an optimal control problem of a stochastic delay equation with state constraints and is treated by the dynamic programming approach.

We first present the study in the simplified case of no delay where a satisfactory theory can be built proving the existence of regular feedback control strategies and then go to the more general case showing some first results on the value function and on its properties.

I will prove that certain pairs of ordered structures are dependent. There are basically two cases depending on whether the smaller structure is dense or discrete. I will discuss the proofs of two quite general theorems which construe the dividing line between these cases. Among examples are dense pairs of o-minimal structures in the first case, and tame pairs of o-minimal structures in the latter. This is joint work with P. Hieronymi.

Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric

modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.

Fibrations are a valuable tool in the study of the geometry of higher dimensional algebraic varieties. By expressing a higher dimensional variety as a fibration by lower dimensional varieties, we can deduce much about its properties. Whilst the theory of elliptic fibrations is very well developed, fibrations by higher dimensional varieties, especially K3 surfaces, are only just beginning to be studied. In this talk I study a special case of the K3-fibration, where the general fibres admit a <2>-polarisation and the base of the fibration is a nonsingular curve.

We study the Neumann and regularity boundary value problems for a divergence form elliptic equation in the plane. We assume the gradient

of the coefficient matrix satisfies a Carleson measure condition and consider data in L^p, 1