Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Spring 2016

date	speaker	title	host(s)
January 22
January 28 (Th 4pm VV901)	Steven Sivek (Princeton)	The augmentation category of a Legendrian knot	Ellenberg
January 29	Ana Caraiani (Princeton)	Locally symmetric spaces, torsion classes, and the geometry of period domains	Ellenberg
February 5	Takis Souganidis (University of Chicago)	Scalar Conservation Laws with Rough Dependence	Lin
February 12	Gautam Iyer (CMU)		Jean-Luc
February 19	Jean-François Lafont (Ohio State)		Dymarz
February 26	Hiroyoshi Mitake (Hiroshima university)		Tran
March 4	Guillaume Bal (Columbia University)		Li, Jin
March 11	Mitchell Luskin (University of Minnesota)	Mathematical Modeling of Incommensurate 2D Materials	Li
March 18	Ralf Spatzier (University of Michigan)	TBA	Dymarz
March 25	Spring Break
April 1	Chuu-Lian Terng (UC Irvine) -->	TBA -->	Mari-Beffa
April 8	Alexandru Ionescu (Princeton)	TBA	Wainger/Seeger
April 15	Igor Wigman (King's College - London)	Nodal Domains of Eigenfunctions	Gurevich/Marshall
April 22	Paul Bourgade (NYU)	TBA	Seppalainen/Valko
April 29	Randall Kamien (U Penn)	TBA	Spagnolie
May 6	Julius Shaneson (University of Pennsylvania)	TBA	Maxim/Kjuchukova

Abstracts

January 28: Steven Sivek (Princeton)

Title: The augmentation category of a Legendrian knot

Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.

The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially. I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.

January 29: Ana Caraiani (Princeton)

Title: Locally symmetric spaces, torsion classes, and the geometry of period domains

Abstract: The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.

I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.

February 5: Takis Souganidis (University of Chicago)

Title: Scalar Conservation Laws with Rough Dependence

I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.

March 11: Mitchell Luskin (UMN)

Title: Mathematical Modeling of Incommensurate 2D Materials

Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.

Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.