# MINT Distinguished Lectures

### Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

### Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

### Abstracts:

#### Can you hear the shape of a drum and deformational spectral rigidity of planar domain

M. Kac popularized the question Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega$ and the associated Dirichlet problem $\Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.$ The set of $\lambda$’s such that this equation has a solution, is called the Laplace spectrum of $\Omega$. Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

Consider the billiard problem inside $\Omega$. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any axis symmetric planar domain with sufficiently smooth boundary close to the disk is dynamically spectrally rigid, i.e. can’t be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.

This is a joint work with J. De Simoi and Q. Wei.

#### Birkhoff Conjecture for convex planar billiards

G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture - namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse.

This is based on several papers with Avila, De Simoi, G.Huang, Sorrentino.

### Carlos Kenig (University of Chicago)

Date Day Time Location Title
2017-03-27 Mon 12:15 Schreiber 006 The energy critical wave equation: An overview
2017-03-29 Wed 14:10 Schreiber 309 The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

### Carlos Kenig (University of Chicago)

Date Day Time Location Title
2017-03-27 Mon 12:15 Schreiber 006 The energy critical wave equation: An overview
2017-03-29 Wed 14:10 Schreiber 309 The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

### Abstracts:

#### The energy critical wave equation: An overview

We will describe the progress in the last 12 years on the understanding of the long-time dynamics for large solutions in the focusing case. Issues like blow-up, scattering and asymptotic decomposition into solitary waves will be discussed.

#### The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

We will discuss the recent work of Duyckaerts-Jia-Kenig-Merle establishing soliton resolution along well chosen sequences of times converging to the final time of existence, for non-radial solutions of the energy critical nonlinear wave equation which remain bounded in the energy space.

### Paul Biran (ETH Zurich)

Date Day Time Location Title
2017-01-09 Mon 12:15 Schreiber 006 A Geometric Outlook on Fukaya Categories
2017-01-11 Wed 14:10 Schreiber 309 It takes Energy to Split Lagrangians

### Paul Biran (ETH Zurich)

Date Day Time Location Title
2017-01-09 Mon 12:15 Schreiber 006 A Geometric Outlook on Fukaya Categories
2017-01-11 Wed 14:10 Schreiber 309 It takes Energy to Split Lagrangians

### Abstracts:

#### A Geometric Outlook on Fukaya Categories

We will explain how the concept of Lagrangian cobordism can be used to approach in a geometric way the Fukaya category of a symplectic manifold (a rather abstract, yet central, object of study in modern symplectic topology). We will also explain how to perform geometric measurements in the Fukaya category via cobordisms. The talk is based on a series of joint works with Octav Cornea.

#### It takes Energy to Split Lagrangians

We will discuss a Hofer Geometry analog that naturally arises from Lagrangian cobordism theory, and see how it can be used to measure relations between Lagrangian submanifolds.

### Nikolai Nadirashvili (Aix-Marseille University)

Date Day Time Location Title
2016-12-12 Mon 12:15 Schreiber 006 Inequalities for eigenvalues of Laplacian on surfaces
2016-12-14 Wed 14:10 Schreiber 309 Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

### Nikolai Nadirashvili (Aix-Marseille University)

Date Day Time Location Title
2016-12-12 Mon 12:15 Schreiber 006 Inequalities for eigenvalues of Laplacian on surfaces
2016-12-14 Wed 14:10 Schreiber 309 Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

### Abstracts:

#### Inequalities for eigenvalues of Laplacian on surfaces

We discuss isoperimetric inequalities for eigenvalues of the Laplacian on two-dimensional Riemannian manifolds. We survey known results, starting with classical inequalities of Hersch and Li–Yau for the first nonzero eigenvalue on a sphere and a projective plane and their recent generalization on the higher eigenvalues.

#### Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

Abstract: We discuss Liouville theorems and some geometric properties of stationary flows of the ideal fluid in dimensions 2 and 3.

### Gordon Slade (University of British Columbia)

Date Day Time Location Title
2016-12-05 Mon 12:15 Schreiber 006 Critical phenomena and renormalisation group
2016-12-05 Mon 14:15 Schreiber 006 Critical phenomena and renormalisation group - in more detail

### Gordon Slade (University of British Columbia)

Date Day Time Location Title
2016-12-05 Mon 12:15 Schreiber 006 Critical phenomena and renormalisation group
2016-12-05 Mon 14:15 Schreiber 006 Critical phenomena and renormalisation group - in more detail

### Abstract:

#### Critical phenomena and renormalisation group

The subject of phase transitions and critical phenomena in statistical mechanics is a rich source of interesting and difficult mathematical problems. There has been considerable success in solving such problems for systems in spatial dimension 2, or in high dimensions, but not in dimension 3. This lecture is intended to provide an introduction to recent work that employs a renormalisation group method to study spin systems and self-avoiding walk in dimension 4 (joint with Bauerschmidt and Brydges), as well as long-range versions of these models in dimensions 1, 2, 3 via an “epsilon expansion”.

### David Conlon (University of Oxford)

Date Day Time Location Title
2016-03-20 Sun 10:05 Schreiber 309 Finite reflection groups and graph norms
2016-03-21 Mon 12:15 Schreiber 006 Recent developments in graph Ramsey theory

### David Conlon (University of Oxford)

Date Day Time Location Title
2016-03-20 Sun 10:05 Schreiber 309 Finite reflection groups and graph norms
2016-03-21 Mon 12:15 Schreiber 006 Recent developments in graph Ramsey theory

### Abstracts:

#### Finite reflection groups and graph norms

Given a graph $H$ and a symmetric function $f:[0,1]^2\rightarrow\mathbb{R}$, define the absolute $H$-norm of $f$ by

where $μ$ is the Lebesgue measure on $[0,1]$. We say that $H$ is weakly norming if the absolute $H$-norm is a norm on the vector space of symmetric functions. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, it is known that even cycles, complete bipartite graphs and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we demonstrate that any graph which is edge-transitive under the action of a certain natural family of automorphisms is weakly norming. This result includes all previous examples of weakly norming graphs and adds many more.

Joint work with Joonkyung Lee.

#### Recent developments in graph Ramsey theory

Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.

In this talk, I will describe some of this progress.

### Rick Kenyon (Brown University)

Date Day Time Location Title
2015-11-15 Sun 10:00 Schreiber 309 A Variational Principle For Permutations
2015-11-16 Mon 12:15 Schreiber 006 Fixed-Energy Harmonic Functions

### Rick Kenyon (Brown University)

Date Day Time Location Title
2015-11-15 Sun 10:00 Schreiber 309 A Variational Principle For Permutations
2015-11-16 Mon 12:15 Schreiber 006 Fixed-Energy Harmonic Functions

### Abstracts:

#### A variational principle for permutations

We study scaling limits of large permutations (permutons’’), constrained by fixing certain of pattern densities, for example, the density of inversions or 123’ patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed 12 density, with fixed 12 and 123 densities, with fixed 12 density and the sum of 123 and 213 densities. In some cases one can find interesting phase transitions as one varies the densities. To obtain our results, we also provide a description of permutons using a dynamic construction.

This is joint work with Daniel Kral, Charles Radin and Pete Winkler.

#### Fixed-energy harmonic functions

We study the map from conductances to edge energies for harmonic functions on graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies.

For rational energies and boundary data the Galois group of $\mathbb{Q}^{tr}$ (the totally real algebraic numbers) over $\mathbb{Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. For planar graphs one can associate tilings of planar regions with rectangles of prescribed areas. Connections with square ice and SLE_{12} (based on work with Angel, Miller, Sheffield, Wilson) will be briefly discussed.

This is joint work with Aaron Abrams.

### Shmuel Weinberger (University of Chicago)

Date Day Time Location Title
2015-11-09 Mon 12:15 Schreiber 006 Topological Problems of Data Analysis, I
2015-11-11 Wed 14:00 Schreiber 209 Topological Problems of Data Analysis, II

### Shmuel Weinberger (University of Chicago)

Date Day Time Location Title
2015-11-09 Mon 12:15 Schreiber 006 Topological Problems of Data Analysis, I
2015-11-11 Wed 14:00 Schreiber 209 Topological Problems of Data Analysis, II

### Abstract

#### Topological Problems of Data Analysis, I, II.

Abstract: Dealing with large and multidimensional data is a ubiquitous problem in many scientific and financial settings. In recent years, geometric and topological methods have become increasingly popular. They suggest problems that are at the interface of probability, differential geometry, topology, and analysis. The first lecture will be devoted to an overview of these ideas, while the second will focus on some topological problems that arise in trying to make some heuristics rigorous and/or practical.

### Van Vu (Yale University)

Date Day Time Location Title
2015-06-08 Mon 12:15 Schreiber 006 Random matrices: Gaps between eigenvalues
2015-06-09 Tue 14:10 Schreiber 209 Roots of random polynomials with general coefficients

### Van Vu (Yale University)

Date Day Time Location Title
2015-06-08 Mon 12:15 Schreiber 006 Random matrices: Gaps between eigenvalues
2015-06-09 Tue 14:10 Schreiber 209 Roots of random polynomials with general coefficients

### Abstracts

#### Random matrices: Gaps between eigenvalues

Gaps (or spacings) between consecutive eigenvalues of a random hermitian matrix plays a central role in random matrix theory. When Wigner used random matrix as a model in physics, it was the behavior of the gaps that captured his attention. Another famous example is the Dyson-Montgomery discussion connecting these gaps with gaps between zeroes of the zeta function.

On the other hand, we still do not understand these gaps very well at the microscopic level. For instance, it has only been proved very recently by Tao and the speaker that for a random +-1 matrix, with high probability all gaps are positive.

In this talk, we will discuss some recent progress in bounding these gaps, based on purely combinatorial techniques, for very general classes of random matrices. We will also present applications in both computer science and numerical analysis (among others, solutions to questions by Babai, and by Deker-Lee-Linial).

Joint work with H. Nguyen (OSU) and T. Tao (UCLA).

#### Roots of random polynomials with general coefficients.

Roots of polynomials with random coefficients have been studied for a long time, starting with fundamental works of Littlewood and Offord in the 1940s, who showed that random polynomials with iid coefficients have very few real roots. Polynomials with iid coefficients are frequently referred to as Kac polynomials.

The number of real roots of Kac polynomials has been computed asymptotically thanks to a remarkable sequence of works by Kac, Erdos-Offord, and Ibragimov-Maslova. However, we know very little about all other cases, where the coefficients are independent, but can have different distributions (different variances, for instance).

In this talk, we are going to give a review of these classical works, focusing on the main ideas behind them, and then present a recent development that provides a good understanding for polynomials with general coefficients with moderate growth (these coefficients can even have implicit variances). As a corollary, we refine and extend Ibragimov-Maslova result for hyperbolic polynomials. The talk is self-contained and several open questions will also be discussed.

Joint work with O. Nguyen (Yale) and Y. Do (UoV).

### Grigori Olshanski (IITP and Higher School of Economics, Moscow)

Date Day Time Location Title
2015-04-14 Tue 14:10 Schreiber 209 Markov processes of algebraic origin
2015-04-15 Wed 14:10 Schreiber 309 What are infinite random permutations?

### Grigori Olshanski (IITP and Higher School of Economics, Moscow)

Date Day Time Location Title
2015-04-14 Tue 14:10 Schreiber 209 Markov processes of algebraic origin
2015-04-15 Wed 14:10 Schreiber 309 What are infinite random permutations?

### Abstracts:

#### Markov processes of algebraic origin

The Brownian motion on the group U(N) of unitary matrices gives rise to a Markov process on the matrix eigenvalues, that is, on N-particle configurations on the circle (Freeman Dyson, 1962). Heuristic arguments show that there should exist an analogous process with infinitely many particles. However, its rigorous construction is a difficult problem, and success in this direction has been achieved only in recent years. I will describe a different model of continuous time Markov dynamics, which is dual to Dyson’s model in the sense that it is formulated in terms of representations of the groups U(N) rather than eigenvalues. It turns out that in the “dual” model, the transition to N=infinity can be carried out by essentially algebraic tools. This is in contrast with Dyson’s model whose study in the N=infinity case requires hard analytic work.

#### What are infinite random permutations?

Random permutations can be viewed as a combinatorial analog of random matrices. In random matrix theory, people study the asymptotic behavior of spectra of large random matrices. Likewise, the literature in combinatorial probability contains many works on the limiting behavior of various characteristics of random permutations of large size. But are there reasonable models of random permutations of actually infinite size? I will describe some positive results in this direction.

### Alex Furman (University of Illinois at Chicago)

Date Day Time Location Title
2015-01-12 Mon 12:15 Schreiber 006 Rigidity for groups with hidden symmetries
2015-01-14 Wed 14:10 Schreiber 007 Simplicity of the Lyapunov spectrum via boundary theory

### Alex Furman (University of Illinois at Chicago)

Date Day Time Location Title
2015-01-12 Mon 12:15 Schreiber 006 Rigidity for groups with hidden symmetries
2015-01-14 Wed 14:10 Schreiber 007 Simplicity of the Lyapunov spectrum via boundary theory

### Abstracts:

#### Simplicity of the Lyapunov spectrum via boundary theory

Consider products of matrices that are chosen using some ergodic stationary random process on $G=SL(d,R)$, e.g. a random walk on $G$. The Multiplicative Ergodic Theorem (Oseledets) asserts that asymptotically such products behave as $\exp(n\Lambda)$ where $\Lambda$ is a fixed diagonal traceless matrix, called the Lyapunov spectrum of the system. The spectrum $\Lambda$ depends on the system in a mysterious way, and is almost never known explicitly. The best understood case is that of random walks, where by the work of Furstenberg, Guivarch-Raugi, and Gol’dsheid-Margulis we know that the spectrum is simple (i.e. all values are distinct) provided the random walk is not trapped in a proper algebraic subgroup. Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich that asserts simplicity of the Lyapunov spectrum for another system related to Teichmuller flow.

In the talk we shall describe an approach to proving simplicity of the spectrum based on ideas from boundary theory that were developed to prove rigidity of lattices.

Based on joint work with Uri Bader.

#### Rigidity for groups with hidden symmetries

Abstract: In the 1970s G.A.Margulis proved that linear representations of certain discrete subgroups (namely lattices) in such Lie groups as SL(3,R), are essentially determined by the ambient Lie group. This phenomenon, known as superrigidity, has far reaching applications and has inspired a large body of research in such areas as geometry, dynamics, descriptive set theory, operator algebras etc.

We shall try to explain the superrigidity of lattices and related groups by looking at some hidden symmetries (Weyl group) that they inherit from the ambient group.

The talk is based on a joint work with Uri Bader.

### Henri Berestycki (EHESS, Paris)

Date Day Time Location Title
2014-12-01 Mon 12:15 Schreiber 006 Propagation in non homogeneous media and applications
2014-12-03 Wed 14:10 Schreiber 209 The effect of domain shape on propagation and blocking for reaction-diffusion equations

### Henri Berestycki (EHESS, Paris)

Date Day Time Location Title
2014-12-01 Mon 12:15 Schreiber 006 Propagation in non homogeneous media and applications
2014-12-03 Wed 14:10 Schreiber 209 The effect of domain shape on propagation and blocking for reaction-diffusion equations

### Abstracts:

#### Propagation in non homogeneous media and applications

The classical theory of the Fisher and Kolmogorov-Petrovsky-Piskunov equation derives the spreading properties for a reaction-diffusion equation in a homogeneous setting. A well known /invasion speed/ governs the asymptotic speed of propagation. This equation plays an important role in a variety of contexts in ecology, biology and physics. In this talk, I will first introduce reaction-diffusion equations, and describe some of their motivations as well as the underlying mechanism. I will then review the classical theory for homogeneous Fisher-KPP equations.

In applications, one often wishes to understand the effects of heterogeneity. This leads to challenging mathematical questions and I will describe some of them. I will then discuss more in detail the effect of inclusion of a line with fast diffusion on biological invasions in the plane. I will report on recent results on this question.

#### The effect of domain shape on propagation and blocking for reaction-diffusion equations

I will discuss reaction-diffusion equations motivated by biology and medicine for which the aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves.

I will first describe the motivations of these questions. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve qualitative results for some non-linear elliptic and parabolic partial differential equations.