# Colloquia/Spring2016

## Spring 2016

## 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.

### March 18: Ralf Spatzier (UMichigan)

CANCELED: Rigidity in Geometry and Dynamics

I will survey some rigidity phenomena in dynamics and also geometry, with emphasis on the notion of higher rank. This first emerged in Margulis’ celebrated work on superrrigidity but has also been important in more recent work on symmetry in dynamical systems. How special is it for maps commute with each other? Smale asked this problem fifty years ago, and answers are finally emerging. Much depends on the differentiability of the maps: it gets harder the more differentiable the map is. Sometimes we can even classify such maps. I’ll discuss this and related phenomena.

### April 8: Alexandru Ionescu (Princeton)

Title: On long-term existence of solutions of water wave models

I will talk about some recent work on long-term/global regularity of solutions of water wave models in 2 and 3 dimensions. The models we consider describe the evolution of an inviscid perfect fluid in a free boundary domain, under the influence of gravity and/or surface tension. This is joint work with Fabio Pusateri and, in part, with Yu Deng and Benoit Pausader.

### April 15: Igor Wigman (KC London)

Title: Topologies of nodal sets of random band limited functions

We show that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions will be determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

The talk should be understood to any advanced undergraduate student.

### April 22: Paul Bourgade (NYU)

Title: Random matrices beyond mean-field

Random matrix statistics were proposed by Eugene Wigner as a new class of universal statistical laws for highly correlated systems.
We will first review established instances of this conjecture for mean-field matrix models. We will then propose an approach towards the spectral analysis of non mean-field models, which are closer to Wigner's original vision.
A key role is played by a new patching of quantum unique ergodicity estimates.

### April 29: Randall Kamien (U Penn)

Title: Liquid Crystals and their (Algebraic) Topology

Liquid Crystals, the materials in your iPhone, are complex materials with varying degrees of internal order. I will discuss and demonstrate how algebraic topology can be used to identify and characterize long-lived configurations. I will also describe how conic sections naturally arise in these structures as intersections of simple polynomials.

### May 4: Peter Sarnak (Princeton and IAS)

Title: Strong approximation for Markoff surfaces

We discuss the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated in part by the finite orbits of these actions on the algebraic points. The latter can be determined effectively and in special cases is connected to the problem of determining all algebraic Painleve VI's. Applications to forms of strong approximation for integer points and to sieving on such affine surfaces, as well as to Markoff numbers will be given.

Joint work with J.Bourgain and A.Gamburd.