## State of the New Proof Assistant

Daniel Grayson
University of Illinois at Urbana-Champaign; Member, School of Mathematics
January 17, 2013 - 11:00am

## Simplicial Types

Peter Lumsdaine
Dalhousie University; Member, School of Mathematics
January 16, 2013 - 11:00am

## Dispersive Estimates for Schroedinger's Equation with a Time-Dependent Potential

Marius Beceanu
Rutgers, The State University of New Jersey; Member, School of Mathematics
January 15, 2013 - 3:15pm

I present some new dispersive estimates for Schroedinger's equation with a time-dependent potential, together with applications.

## Faster Numerical Linear Algebra Algorithms Via Sparser Subspace Embeddings

Jelani Nelson
Member, School of Mathematics, IAS
January 15, 2013 - 10:30am

## On Bilinear Complexity

Pavel Hrubes
University of Washington
January 14, 2013 - 11:15am

For a set of polynomials F, we define their bilinear complexity as the smallest k so that F lies in an ideal generated by k bilinear polynomials. The main open problem is to estimate the bilinear complexity of the single polynomial $\sum_{i,j}x_i^2 y_j^2$. This question is related to the classical sum-of-squares problem as well as to problems in arithmetic circuit complexity. We will focus on related sets of polynomials and prove some lower and upper bounds on their bilinear complexity.

## The SOS (aka Lassere/Positivestellensatz/Sum-of-Squares) System Series

Raghu Meka (1) and Avi Wigderson (2)
DIMACS (1) and Professor, School of Mathematics, IAS (2)
December 18, 2012 - 10:30am

We will give an overview of this system, which has been at the center of recent algorithmic and proof complexity developments. We will give the definitions of the system (as a proof system for polynomial inequalities, and as an SDP-based algorithm), and basic upper and lower bounds for it. In particular we'll explain the recent SOS-proof of the hypercontractive inequality for the noisy hypercube of Barak et al., as well as the degree lower bounds for proving Tseitin and Knapsack tautologies of Grigoriev.

## Local Global Principles for Galois Cohomology

Julia Hartmann
RWTH Aachen University; Member, School of Mathematics, IAS
December 13, 2012 - 4:30pm

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field.
Motivated by work of Kato and others for n=3, we show that local-global principles hold for
H^n(F, Z/mZ(n-1)) for all n>1.
In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H^1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G.

## Invariance Under Isomorphism and Definability

Per Martin-Löf
Stockholm University, Member, School of Mathematics, IAS
December 13, 2012 - 11:00am

## Universality in Mean Curvature Flow Neckpinches

Gang Zhou
University of Illinois at Urbana-Champaign
December 12, 2012 - 3:30pm

This is from joint works with D. Knopf and I. M. Sigal. In this talk I will present a new strategy in studying neckpinching of mean curvature flow. Different from previous results, we do not use backward heat kernel, entropy estimates or subsequent convergence, instead we apply almost precise estimates, invented in the past few years, to obtain the first result on asymmetric surface.

Michael Shulman