Members Seminar

Algebraic combinatorics: applications to statistical mechanics and complexity theory

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.

Everything you wanted to know about machine learning but didn't know whom to ask

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.

Representations of Kauffman bracket skein algebras of a surface

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.

Decomposition theorem for semisimple algebraic holonomic D-modules

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.

Weinstein manifolds through skeletal topology

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.

High density phases of hard-core lattice particle systems

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.

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.