On Sofic Groups
Abstract
The class of sofic groups was introduced by Gromov in 1999. It
includes all residually finite and all amenable groups. In fact, no group has been proved
not to be sofic, so it remains possible that all groups are sofic. Their
defining property is that, roughly speaking, for any finite subset F of
the group G, there is a map from G to a finite symmetric group, which is
approximates to an injective homomorphism on F. The widespread interest in
these group results partly from their connections with other branches of
mathematics, including dynamical systems. In the talk, we will concentrate
on their definition and algebraic properties.
Option pricing, fake Brownian motion, and minimal variation
Abstract
Suppose we are given a double continuum (in time and strike) of discounted
option prices, or equivalently a set of measures which is increasing in
convex order. Given sufficient regularity, Dupire showed how to construct
a time-inhomogeneous martingale diffusion which is consistent with those
prices. But are there other martingales with the same 1-marginals? (In the
case of Gaussian marginals this is the fake Brownian motion problem.)
In this talk we show that the answer to the question above is yes.
Amongst the class of martingales with a given set of marginals we
construct the process with smallest possible expected total variation.
Limit method in extremal combinatorics
Abstract
Razborov's flag algebras provide a formal system
for operating with asymptotic inequalities between subgraph densities,
allowing to do extensive "book-keeping" by a computer. This novel use
of computers led to progress on many old problems of extremal
combinatorics. In some cases, finer structural information can be
derived from a flag algebra proof by by using the Removal Lemma or
graph limits. This talk will overview this approach.
Coarse median spaces
Abstract
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.
13:00
Cayley graphs of Fuchsian surface groups versus hyperbolic graphs
Abstract
Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.
Shimura Decomposition and Tunnell-like formulae.
Abstract
Let k be an odd integer and N be a positive integer divisible by 4. Let g be a newform of weight k - 1, level dividing N/2 and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight k/2 that are 'Shimura-equivalent' to g. Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.
Euler systems for Rankin--Selberg convolutions of modular forms
Abstract
I will describe a construction of special cohomology classes over the cyclotomic tower for the product of the Galois representations attached to two modular forms, which $p$-adically interpolate the "Beilinson--Flach elements" of Bertolini, Darmon and Rotger. I will also describe some applications to the Iwasawa theory of modular forms over imaginary quadratic fields.