School of Mathematics

CSDM: A Tutorial on the Likely Worst-Case Complexities of NP-Complete Problems

Russell Impagliazzo
Institute for Advanced Study
January 24, 2012
Abstract
The P vs. NP problem has sometimes been unofficially paraphrased as asking whether
it is possible to improve on exhaustive search for such problems as Satisfiability, Clique,
Graph Coloring, etc. However, known algorithms for each of these problems indeed are
substantially better than exhaustive search, if still exponential. Furthermore, although a
polynomial-time algorithm for any one of these problems implies one for all of them, these
improved exponential algorithms are highly specific, and it is unclear what the limit of
improvement should be.

Members Seminar: The Role of Symmetry in Phase Transitions

Tom Spencer
Professor, School of Mathematics, Institute for Advanced Study
January 23, 2012

This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.

Around the Davenport-Heilbronn Function

Enrico Bombieri
Institute for Advanced Study
November 10, 2011

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In this lecture, we study the Moebius inversion for functions of this type and show how its behavior is related to the distribution of zeros in the half-plane of absolute convergence. Work in collaboration with Amit Ghosh.

Vertex Sparsification: An Introduction, Connections and Applications

Ankur Moitra
Massachusetts Institute of Technology; Institute for Advanced Study
November 8, 2011

The notion of exactly (or approximately) representing certain combinatorial properties of a graph $G$ on a simpler graph is ubiquitous in combinatorial optimization. In this talk, I will introduce the notion of vertex sparsification. Here we are given a graph $G = (V, E)$ and a set of terminals $K \subset V$ and our goal is to find one single graph $H = (K, E_H)$ on just the terminal set so that $H$ approximately preserves the minimum cut between every bi-partition of the terminals.

Strong and Weak Epsilon Nets and Their Applications

Noga Alon
Tel Aviv University; Institute for Advanced Study
November 7, 2011

I will describe the notions of strong and weak epsilon nets in range spaces, and explain briefly some of their many applications in Discrete Geometry and Combinatorics, focusing on several recent results in the investigation of the extremal questions that arise in the area, and mentioning some of the remaining open problems.