Quantum Fingerprints that Keep Secrets

Dmitry Gavinsky
NEC Research Laboratories
April 18, 2011

In a joint work with Tsuyoshi Ito we have constructed a fingerprinting scheme (i.e., hashing) that leaks significantly less than log(1/epsilon) bits about the preimage, where epsilon is the error ("collision") probability. It is easy to see that classically this is not achievable; our construction is quantum, and it gives a new example of (unconditional) qualitative advantage of quantum computers.

New Tools for Graph Coloring

Rong Ge
Princeton University
April 19, 2011

How to color $3$ colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time algorithm uses $n^{0.2130}$ colors.

We explore the possibility that more levels of Lasserre Hierarchy can give improvements over previous algorithms. While the case of general graphs is still open, we can give analyse the Lasserre relaxation for two interesting families of graphs.

Universality in the 2D Coulomb Gas

Pierluigi Falco
Member, School of Mathematics
April 20, 2011

The Coulomb Gas is a model of Statistical Mechanics with a special type of phase transition. In the first part of the talk I will review the expected features conjectured by physicists and the few mathematical results so far obtained. The second part will be an introductory discussion of a general technique (Renormalization Group) to approach the problem.

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) .

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.

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.