Cambridge

Let $A$ be a finite set in some ambient group. We say that $A$ is a $K$-approximate group if $A$ is symmetric and if the set $A.A$ (the set of all $xy$, where $x$, $y$ lie in $A$) is covered by $K$ translates of $A$. I will illustrate this notion by example, and will go on to discuss progress on the "rough classification" of approximate groups in various settings: abelian groups, nilpotent groups and matrix groups of fixed dimension. Joint work with E. Breuillard.

I will describe joint work with Mohammed Abouzaid, in which we complete the proof of homological mirror symmetry for the standard four-torus and consider various applications. A key tool is the recently-developed holomorphic quilt theory of Mau-Wehrheim-Woodward.

Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way.  Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.

I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold.  A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained.  This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.

All necessary tools from stochastic calculus and geometry will be explained.

We consider the classical problem of forming portfolios of vanilla options in order to hedge more exotic derivatives. In particular, we focus on a model in which the agent can trade a stock and a family of variance swaps written on that stock. The market is only approximately complete in the sense that any submarket consisting of the stock and the variance swaps of a finite set of maturities is incomplete, yet every bounded claim is in the closure of the set of attainable claims. Taking a Hilbert space approach, we give a characterization of hedging portfolios for a certain class of contingent claims. (Joint work with Francois Berrier)