Before the "geometric Satake equivalence" there was a decategorified version of it which however contained most of its essential features. In my talk I will talk about some of the ideas which have led to this theory. In particular I will explain the...

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School of Mathematics

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n)Sym(n) (in the sofic case) or the finite dimensional...

I will discuss techniques, structural results, and open problems from two of my recent papers, in the context of a broader area of work on the motivating question: "how do we get the most from our data?"

In the first part of the talk, I will...

For the incompressible Navier-Stokes equations, classical results state that weak solutions are unique in the so-called Ladyzhenskaya-Prodi-Serrin regime. A scaling analysis suggests that classical uniqueness results are sharp, but current...

We present an overview of elementary methods to study extensions of modular representations of various types of "groups". We shall begin by discussing actions of an elementary abelian pp-group, E=(Z/p)rE=(Z/p)r, on finite dimensional vector spaces...

What combinatorial properties are likely to be satisfied by a random subspace over a finite field? For example, is it likely that not too many points lie in any Hamming ball of fixed radius? What about any combinatorial rectangle of fixed side...

### Approximations of groups, subquotients of infinite direct products and equations over groups

Let C be a class of groups. (For example, C is a class of all finite groups, or C is a class of all finite symmetric groups.) I give a definition of approximations of a group G by groups from C. For example, the groups approximable by symmetric...

Let X and Y be normed spaces. In functional analysis, a ``theorem on factorization through L2'' refers to the following type of statement:

Every bounded linear operator A mapping X to Y (i.e. sup_x ||A(x)||_Y/||x||_X

Such factorization...

An extension of Gromov compactness theorem ensures that any family of manifolds with convex boundaries, uniform bound on the dimension and uniform lower bound on the Ricci curvature is precompact in the Gromov-Hausdorff topology. In this talk, we...

In a joint work with Tianyi Zheng we show that the growth function of the first Grigorchuk group satisfies

lnlnvn/lnvn=a,

where a=log2/logx, x being a positive root of the polynomial x3?x2?2x?4. This is done by constructing measures with...