School of Mathematics

A Unified Framework for Testing Linear-Invariant Properties

Arnab Bhattacharyya
Massachusetts Institute of Technology
October 18, 2010

In a sequence of recent papers, Sudan and coauthors have investigated the relation between testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Linear-invariance is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties.

Voting Paradoxes and Combinatorics

Noga Alon
Institute for Advanced Study, Visiting Professor
October 13, 2010

The early work of Condorcet in the eighteenth century, and that of Arrow and others in the twentieth century, revealed the complex and interesting mathematical problems that arise in the theory of social choice. In this lecture, Noga Alon, Visiting Professor in the School of Mathematics, explains how the simple process of voting leads to strikingly counter-intuitive paradoxes, focusing on several recent intriguing examples.

The Complexity of the Non-commutative Determinant

Srikanth Srinivasan
Institute for Advanced Study
October 11, 2010

I will talk about the computational complexity of computing the noncommutative determinant. In contrast to the case of commutative algebras, we know of (virtually) no efficient algorithms to compute the determinant over non-commutative domains. Our results show that the determinant in noncommutative settings can be as hard as the permanent.

Potential Automorphy

Richard Taylor
Institute for Advanced Study
October 4, 2010

I will introduce l-adic representations and what it means for them to be automorphic, talk about potential automorphy as an alternative to automorphy, explain what can currently be proved (but not how) and discuss what seem to me the important open problems. This should serve as an introduction to half the special year for non-number theorists. The other major theme will likely be the `p-adic Langlands program', which I will not address (but perhaps someone else will).