Members Seminar

Geometry and arithmetic of sphere packings

Alex Kontorovich
Rutgers University
October 23, 2017
We introduce the notion of a "crystallographic sphere packing," which generalizes the classical Apollonian circle packing. Tools from arithmetic groups, hyperbolic geometry, and dynamics are used to show that, on one hand, there is an infinite zoo of such objects, while on the other, there are essentially finitely many of these, in all dimensions. No familiarity with any of these topics will be assumed.

Analysis and topology on locally symmetric spaces

Akshay Venkatesh
Stanford University; Distinguished Visiting Professor, School of Mathematics
October 9, 2017
Locally symmetric spaces are a class of Riemannian manifolds which play a special role in number theory. In this talk, I will introduce these spaces through example, and show some of their unusual properties from the point of view of both analysis and topology. I will conclude by discussing their (still very mysterious) relationship with algebraic geometry.

Algebra and geometry of the scattering equations

Peter Goddard
Professor Emeritus, School of Natural Sciences
April 3, 2017
Four years ago, Cachazo, He and Yuan found a system of algebraic equations, now named the "scattering equations", that effectively encoded the kinematics of massless particles in such a way that the scattering amplitudes, the quantities of physical interest, in gauge theories and in gravity could be written as sums of rational functions over their solutions.

Efficient non-convex polynomial optimization and the sum-of-squares hierarchy

David Steurer
Cornell University; Member, School of Mathematics
March 20, 2017

The sum-of-squares (SOS) hierarchy (due to Shor'85, Parrilo'00, and Lasserre'00) is a widely-studied meta-algorithm for (non-convex) polynomial optimization that has its roots in Hilbert's 17th problem about non-negative polynomials.

SOS plays an increasingly important role in theoretical computer science because it affords a new and unifying perspective on the field's most basic question:

What's the best possible polynomial-time algorithm for a given computational problem?

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.

Mirror symmetry via Berkovich geometry I: overview

Tony Yue Yu
Visitor, School of Mathematics
February 13, 2017
Berkovich geometry is an enhancement of classical rigid analytic geometry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds. I will explain (1) what is mirror symmetry, (2) what are Berkovich spaces, (3) how Berkovich spaces appear naturally in the study of mirror symmetry, and (4) how we obtain a better understanding of several aspects of mirror symmetry via this viewpoint. This member seminar serves also as an overview of my minicourses in the same week.

Local systems and the Hofer-Zehnder capacity

Alexandru Oancea
Université Pierre et Marie Curie; Member, School of Mathematics
February 6, 2017
The Hofer-Zehnder capacity of a symplectic manifold is one of the early symplectic invariants: it is a non-negative real number, possibly infinite. Finiteness of this capacity has strong consequences for Hamiltonian dynamics, and it is an old question to decide whether it holds for small compact neighborhoods of closed Lagrangians. I will explain a positive answer to this question for a class of manifolds whose free loop spaces admit nontrivial local systems.

Homological versus Hodge-theoretic mirror symmetry

Timothy Perutz
University of Texas, Austin; von Neumann Fellow, School of Mathematics
January 30, 2017
I'll describe joint work with Sheel Ganatra and Nick Sheridan which rigorously establishes the relationship between different aspects of the mirror symmetry phenomenon for Calabi-Yau manifolds. Homological mirror symmetry---an abstract, categorical statement---implies Hodge theoretic mirror symmetry, a concrete relation between counts of rational curves and variations of Hodge structure.