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.

Edward T. Cone Concert: So Percussion

So Percussion with David Lang
March 3, 2017
The season will conclude with a performance by So Percussion, the Edward T. Cone Ensemble-in-Residence at Princeton University. The concerts will feature a program by composer Steve Reich, in honor of his eightieth birthday. A concert talk with David Lang and the artists will follow the Friday, March 3 performance.

Book Talk: The Invention of Humanity

Nicola di Cosmo, Jonathan Israel, Michael Walzer, and Siep Stuurman
March 1, 2017
During most of history, inequality was the habitual and reasonable standard, while equality stood in need of justification, if it was considered at all. Inequality was omnipresent, palpable and realistic, while equality had to be imagined, argued for, conjured up from somewhere. In short, equality had to be invented. In this public book talk, Siep Stuurman will discuss the themes of his book The Invention of Humanity (Harvard University Press) in a panel discussion with Nicola di Cosmo, Jonathan Israel, and Michael Walzer.

Canonical coordinates for Calabi Yau manifolds II

Sean Keel
University of Texas, Austin; Member, School of Mathematics
March 3, 2017
In the two talks, aimed at a broad mathematical audience, I'll explain my conjecture, joint with Gross, Hacking, and Siebert, which says, informally, that if you live on a Calabi Yau, your world comes with an intrinsic global positioning system. More precisely: that the coordinate ring comes with a canonical vector space basis, such that the structure constants for the multiplication rule are given by counts of rational curves (on the mirror).