Past

Domain decomposition methods have been developed in various contexts, and with very different goals in mind. I will start my presentation with the historical inventions of the Schwarz method, the Schur methods and Waveform Relaxation. I will show for a simple model problem how all these domain decomposition methods function, give precise results for the model problem, and also explain the most general convergence results available currently for these methods. I will conclude with the parareal algorithm as a new variant for parallelization of evolution problems in the time direction.

General Arrow-Debreu equilibrium can be determined for expected utility maximisers by explicit solutions for individual players. When the expected

utilities are distorted by probability weighting functions, players cannot find explicit optimal decisions. Zhou and Xia studied the existence of equilibrium when the probability weighting functions are the same for all individual players. In this paper, we investigate the same problem but with heterogeneous probability weighting function.

I will discuss the motion of screw dislocations in an elastic body under antiplane shear. In this setting, dislocations are viewed as points in a two-dimensional domain where the strain field fails to be a gradient. The motion is determined by the Peach-Koehler force and the slip-planes in the material. This leads to a system of discontinuous ODE, where the vector field depends on the solution to an elliptic PDE with Neumann data. We show short-time existence of solutions; we also have uniqueness for a restricted class of domains. In general, global solutions do not exist because of collisions.

 Following Prestel and Ziegler, we will explore what it means for a field
to be t-henselian, i.e. elementarily equivalent (in the language of
rings) to some non-trivially henselian valued field. We will discuss
well-known as well as some new properties of t-henselian fields.

***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON THURSDAY 31ST JANUARY *****

We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

I will look at some tools for proving expansion in the Cayley graphs of finite quotients of a given infinite group, with particular emphasis on Bourgain-Gamburd’s work on expansion in Zariski-dense subgroups of SL_2(Z), and speculate to what extent such expansion may be said to be “uniform”.

I will introduce the notion of Kurosh rank for subgroups of 
free products. This rank satisfies the Howson property, i.e. the 
intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.
I will present a version of the Strengthened Hanna Neumann inequality in 
the case of free products of right-orderable groups. Joint work with  A. 
Martino and I. Schwabrow.

A self-avoiding walk on a lattice is a walk that never visits the same vertex twice.  Self-avoiding walks (SAW) have attracted interest for decades, first in statistical physics, where they are considered as polymer models, and then in combinatorics and in probability theory (the first mathematical contributions are probably due to John Hammersley, from Oxford, in the early sixties). However, their properties remain poorly understood in low dimension, despite the existence of remarkable conjectures.

About two years ago, Duminil-Copin and Smirnov proved an "old" and remarkable conjecture of Nienhuis (1982), according to which the number of SAWs of length n on the honeycomb (hexagonal) lattice grows like mu^n, with mu=sqrt(2 +sqrt(2)).

This beautiful result has woken up the hope to prove other simple looking conjectures involving these objects. I will thus present the proof of a younger conjecture (1995) by Batchelor and Yung, which deals with SAWs confined to a half-plane and interacting with its boundary.

(joint work with N. Beaton, J. de Gier, H. Duminil-Copin and A. Guttmann)

The talk will discuss generalizations of the classical Hadamard jump  condition to general locally Lipschitz maps, and applications to
polycrystals. This is joint work with Carsten Carstensen.

By a "coarse median" we mean a ternary operation on a path metric space, satisfying certain conditions which generalise those of a median algebra. It can be interpreted as a kind of non-positive curvature condition, and is applicable, for example to finitely generated groups. It is a consequence of work of Behrstock and Minsky, for example, that the mapping class group of a surface satisfies this condition. We aim to give some examples, results and applications concerning this notion.

Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

A multi-dimensional extension of the structural default model with firms' values driven by diffusion processes with Marshall-Olkin-inspired

correlation structure is presented. Semi-analytical methods for solving

the forward calibration problem and backward pricing problem in three

dimensions are developed. The model is used to analyze bilateral counter- party risk for credit default swaps and evaluate the corresponding credit and debt value adjustments.