Computer Science and Discrete Mathematics (CSDM)

Theoretical Computer Science and Discrete Mathematics

CSDM - Twice-Ramanujan Sparsifiers

Nikhil Srivastava
Yake University
September 21, 2009

We prove that every graph has a spectral sparsifier with a number of edges linear in its
number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our
sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs.
In particular, we prove that for every d > 1 and every undirected, weighted graph G =
(V,E,w) on n vertices, there exists a weighted graph H = (V, F, ~w) with at most dn edges
such that for every x ∈ ℜV ,

 

CSDM - On P vs NP, Geometric Complexity Theory, and the Riemann Hypothesis

Ketan Mulmuley
Institute for Advanced Study
February 9, 2009

This series of three talks will give a nontechnical, high level overview of geometric complexity theory (GCT), which is an approach to the P vs. NP problem via algebraic geometry, representation theory, and the theory of a new class of quantum groups, called nonstandard quantum groups, that arise in this approach. In particular, GCT suggests that the P vs. NP problem in characteristic zero is intimately linked to the Riemann Hypothesis over finite fields. No background in algebraic geometry, representation theory or quantum groups would be
assumed.