Colloquia/Fall18

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)	Homogenization and Anomalous Diffusion	Jean-Luc
February 19	Jean-François Lafont (Ohio State)	Rigidity and flexibility of almost-isometries	Dymarz
February 26	Hiroyoshi Mitake (Hiroshima university)	On asymptotic speed of the crystal growth	Tran
March 4	Guillaume Bal (Columbia University)	Inverse and Control Transport Problems	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.

February 12: Gautam Iyer (CMU)

Homogenization and Anomalous Diffusion

Homogenization is a well known technique used to approximate the macroscopic behaviour of a material with microscopic impurities. While this originally arose in the study of composite materials, it has applications to various other fields, and I will focus on a few results motivated by fluid dynamics. One well known result in this direction is by GI Taylor estimating the dispersion rate of a solute in a pipe. The length scales involved in typical pipelines, however, are too short for this result to apply. I will conclude with a few recent "intermediate time" results describing the effective behaviour in scaling regimes outside those of standard homogenization results.

February 19: Jean-François Lafont (Ohio State)

Rigidity and flexibility of almost-isometries

An almost isometry (AI) is a quasi-isometry (QI) with multiplicative constant =1. Given two metrics on a closed manifold, Milnor-Swarc implies that the lifted metrics on the universal cover are QI to each other. When are they AI to each other? In the rigidity direction, we give various examples where the only time such lifts are AI is when they are isometric (joint with Kar and Schmidt). In the flexible direction, we show that for higher genus surfaces, any two metrics have lifts which, after possibly scaling one of the lifted metrics, are AI to each other (joint with Schmidt and van Limbeek). In the latter examples, one can further show that the AI is usually not equivariant with respect to the group actions.

February 26: Hiroyoshi Mitake (Hiroshima university)

In the talk, I will propose a model equation to study the crystal growth as a prototype, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears because of the double nonlinear effects coming from the surface evolution and the source term, which is sensitive to the shapes of source terms. This is a joint work with Y. Giga (U. Tokyo), and H. V. Tran (U. Wisconsin-Madison).

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.

Past Colloquia

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012