"Small" representations of finite classical groups

Shamgar Gurevich
University of Wisconsin and Yale University
March 8, 2017

Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these Fourier type formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the "small" representations tend to contribute the largest terms to these sums, so a systematic knowledge on them might assist in the solution of important problems.

On small sums of roots of unity

Philipp Habegger
University of Basel
March 9, 2017

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most exponentially in $N$ (for fixed $k$). Moreover it is known that the decay is at worst polynomial if $k < 5$. But no general sub-exponential bound is known if $k \geq 5$.

On the role of rank in representation theory of the classical groups

Roger Howe
Yale University and Texas A & M University
March 8, 2017
Rank gives a natural filtration on representations of classical groups. The eta correspondence provides a clean description of a natural family of representations of a given rank, subject to certain bounds. There is evidence that this construction produces all representations of small rank.

Some basic problems and results from Invariant Theory

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics
March 7, 2017

Invariant theory deals with properties of symmetries - actions of groups on sets of objects.

It has been slower to join its sibling mathematical theories in the computer science party, but we now see it more and more in studies of computational complexity, pseudorandomness, and the analysis of algorithms.


Information complexity and applications

Mark Braverman
Princeton University; von Neumann Fellow, School of Mathematics
March 6, 2017

Over the past two decades, information theory has reemerged within computational complexity theory as a mathematical tool for obtaining unconditional lower bounds in a number of models, including streaming algorithms, data structures, and communication complexity. Many of these applications can be systematized and extended via the study of information complexity – which treats information revealed or transmitted as the resource to be conserved.

Interactive coding with nearly optimal round and communication blowup

Yael Kalai
Microsoft Research
March 6, 2017

The problem of constructing error-resilient interactive protocols was introduced in the seminal works of Schulman (FOCS 1992, STOC 1993). These works show how to convert any two-party interactive protocol into one that is resilient to constant-fraction of error, while blowing up the communication by only a constant factor. Since these seminal works, there have been many follow-up works which improve the error rate, the communication rate, and the computational efficiency.