# School of Mathematics

## On Zaremba's Conjecture

Alex Kontorovich
Stony Brook University
April 22, 2011

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

Zeev Dvir
Princeton University; Member, School of Mathematics
April 22, 2011

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

Larry Guth
University of Toronto; Member, School of Mathematics
April 22, 2011

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

Swastik Kopparty
Member, School of Mathematics
April 22, 2011

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) .

## Random Walks in Linear Groups

Peter Varju
Princeton University
April 22, 2011