# School of Mathematics

## Quadratic Goldreich-Levin Theorems

Decompositions in theorems in classical Fourier analysis which decompose a function into large Fourier coefficients and a part that is pseudorandom

## Learning and Testing k-Model Distributions

A k-modal probability distribution over the domain {1,...,N} is one whose histogram has at most k "peaks" and "valleys". Such distributions are a natural generalization of the well-studied class of monotone increasing (or monotone decreasing) probability distributions.

## Pseudorandomness in Mathematics and Computer Science Mini-Workshop

In math, one often studies random aspects of deterministic systems and structures. In CS, one often tries to efficiently create structures and systems with specific random-like properties. Recent work has shown many connections between these two approaches through the concept of "pseudorandomness". This workshop highlights these connections, aimed at a joint audience of mathematicians and computer scientists.

## On Zaremba's Conjecture

Inspired by the theory of good lattice points in numerical integration, Zaremba conjectured in 1972 that for every denominator q, there is some coprime numerator p, such that the continued fraction expansion of p/q has uniformly bounded quotients. We will present recent progress on this problem, joint with Jean Bourgain.

## Monotone Expanders -- Constructions and Applications

A Monotone Expander is an expander graph which can be decomposed into a union of a constant number of monotone matchings, under some fixed ordering of the vertices. A matching is monotone if every two edges (u,v) and (u',v') in it satisfy u < u' --> v < v'. It is not clear a priori if monotone expanders exist or not. This is partially due to the fact that the natural application of the probabilistic method does not work in this special case.

## The Polynomial Method and Applications From Finite Field Kakeya to Distinct Distances

## The Correlation of Multiplicative Characters with Polynomials over Finite Fields

This talk will focus on the complexity of the cubic-residue (and higher-residue) characters over GF(2^n) , in the context of both arithmetic circuits and polynomials.

We show that no subexponential-size, constant-depth arithmetic circuit over GF(2) can correctly compute the cubic-residue character for more than 1/3 + o(1) fraction of the elements of GF(2^n). The key ingredient in the proof is an adaptation of the Razborov-Smolensky method for circuit lower bounds to setting of univariate polynomials over GF(2^n) .