# School of Mathematics

## Weak Infinity Groupoids in HoTT

## The Ribe Program

A linear property of Banach spaces is called "local" if it depends on finite number of vectors and is invariant under renorming (i.e., distorting the norm by a finite factor). A famous theorem of Ribe states that local properties are invariant under (non linear) uniform-homeomorphisms, suggesting that local properties should have purely metric characterizations.

The Ribe program attempts to uncover explicit metric characterizations of local properties, and study them in the context of metric spaces. More broadly it attempts to apply ideas from Banach

## Toeplitz Matrices and Determinants Under the Impetus of the Ising Model

This is the second of two talks in which the speaker will discuss the development of the theory of Toeplitz matrices and determinants in response to questions arising in the analysis of the Ising model of statistical mechanics. The first talk will be largely historical and the second will describe the state of the art today, including recent results of the speaker and his colleagues Alexander Its and Igor Krasovsky.

## New Independent Source Extractors with Exponential Improvement

## Toeplitz Matrices and Determinants Under the Impetus of the Ising Model

This is the first of two talks in which the speaker will discuss the development of the theory of Toeplitz matrices and determinants in response to questions arising in the analysis of the Ising model of statistical mechanics. The first talk will be largely historical and the second will describe the state of the art today, including recent results of the speaker and his colleagues Alexander Its and Igor Krasovsky.

## Orthogonal factorization in HoTT

## Abelian varieties with maximal Galois action on their torsion points

Abstract:

Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)

## Homotopy and Univalence

## Sparsity Lower Bounds for Dimensionality Reducing Maps

Abstract:

We give near-tight lower bounds for the sparsity required in several dimensionality reducing linear maps. In particular, we show:

(1) The sparsity achieved by [Kane-Nelson, SODA 2012] in the sparse Johnson-Lindenstrauss lemma is optimal up to a log(1/eps) factor.

(2) RIP_2 matrices preserving k-space vectors in R^n with the optimal number of rows must be dense as long as k < n / polylog(n).