## Rigidity and recurrence in symplectic dynamics

Matthias Schwarz

Universität Leipzig; Member, School of Mathematics

December 11, 2017

Matthias Schwarz

Universität Leipzig; Member, School of Mathematics

December 11, 2017

Greta Panova

University of Pennsylvania; von Neumann Fellow, School of Mathematics

December 4, 2017

We will give a brief overview of the classical topics, problems and results in Algebraic Combinatorics. Emerging from the representation theory of $S_n$ and $GL_n$, they took a life on their own via the theory of symmetric functions and Young Tableaux and found applications into new fields. In particular, these objects can describe integrable lattice models in statistical mechanics like dimer covers on the hexagonal grid, aka lozenge tilings.

Sanjeev Arora

Princeton University; Visiting Professor, School of Mathematics

November 27, 2017

This talk is going to be an extended and more technical version of my brief public lecture https://www.ias.edu/ideas/2017/arora-zemel-machine-learning

I will present some of the basic ideas of machine learning, focusing on the mathematical formulations. Then I will take audience questions.

Helen Wong

Carleton University; von Neumann Fellow, School of Mathematics

November 20, 2017

The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and a representation of the skein algebra features in Witten's topological quantum field theory interpretation of the Jones invariant. Later, the skein algebra and its representations was discovered to bear deep relationships to hyperbolic geometry, via the $SL_2 \mathbb C$-character variety of the surface.

Takuro Mochizuki

Kyoto University

November 13, 2017

Decomposition theorem for perverse sheaves on algebraic varieties, proved by Beilinson-Bernstein-Deligne-Gabber, is one of the most important and useful theorems in the contemporary mathematics. By the Riemann-Hilbert correspondence, we may regard it as a theorem for regular holonomic D-modules of geometric origin. Rather recently, it was generalized to the context of semisimple holonomic D-modules which are not necessarily regular.

Laura Starkston

Stanford University

October 30, 2017

We will discuss how to study the symplectic geometry of $2n$-dimensional Weinstein manifolds via the topology of a core $n$-dimensional complex called the skeleton. We show that the Weinstein structure can be homotoped to admit a skeleton with a unique symplectic neighborhood. Then we further work to reduce the remaining singularities to a simple combinatorial list coinciding with Nadler's arboreal singularities.

Ian Jauslin

Member, School of Mathematics

October 30, 2017

In this talk, I will discuss the behavior of hard-core lattice particle systems at high fugacities. I will first present a collection of models in which the high fugacity phase can be understood by expanding in powers of the inverse of the fugacity. I will then discuss a model in which this expansion diverges, but which can still be solved by expanding in other high fugacity variables. This model is an interacting dimer model, introduced by O.Heilmann and E.H.Lieb in 1979 as an example of a nematic liquid crystal.

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.

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.

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.