# Seminars & Events for Analysis Seminar

##### A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line

In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse. The goal of the present work is to construct an asymptotic global-in-time modulated two-soliton solution of small mass, which exhibits the following two regimes: (i) a turbulent regime characterized by an explicit growth of high Sobolev norms on a finite time interval, followed by (ii) a stabilized regime in which the high Sobolev norms remain stationary large forever in time. This talk is based on joint work with P. Gerard (Orsay, France), E. Lenzmann (Basel, Switzerland), and P. Raphael (Nice, France).

##### Focal points and sup-norms of eigenfunctions

**Please note special location. **If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x_0\in M for the geodesic flow at which the associated Perron-Frobenius operator U: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase. In two dimensions, the condition simplifies and is equivalent to the condition that there be a point through which the geodesic flow is periodic. This is joint work with Steve Zelditch.

##### A geometric approach for sharp Local well-posedness of quasilinear wave equations

**Please note special location and time. **The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by him and Rodnianski. Via this approach they obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, $H^{2+\epsilon}$, local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations.

##### On two extremal problems for the Fourier transform

One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps $L^p$ boundedly to $L^q$, where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space, the optimal constant in this inequality was found by Babenko for $q$ an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality.

##### Remarks on Prandtl boundary layers

In this talk we will review some recent results on the instability of Prandtl boundary layers which arise in the inviscid limit of incompressible Navier Stokes equations near a boundary (joint work with Y. Guo and T. Nguyen).

##### Decouplings and applications

We describe a new Fourier analytic method for estimating a wide variety of exponential sums. The talk will mainly focus on the applications to number theory and PDEs. This is joint work with Jean Bourgain.

##### TBA - Nahmod: CANCELLED - NEW DATE TBD

##### A free boundary problem in kinetic theory

We consider a rigid body colliding with a continuum of particles. We assume that the body is moving at a velocity close to an equilibrium velocity V_{infty} and that the particles colliding with the body reflect probabilistically with some probablility distribution K. The fact that the particles and the body might collide many times or even infinitely many times makes the problem highly nontrivial even we only consider simple cases. We prove that the system does tend to an equilibrium. Moreover, we find a condition that is sufficient and almost necessary that the collective force of the colliding particles reverses the relative velocity V(t) of the body, that is, changes the sign of V(t)-V_{infty}, before the body approaches equilibrium. Examples of both reversal and irreversal are given.

##### TBA - Hintz

##### From Newton's dynamics to the heat equation

**This is a joint Analysis - Analysis of Fluids and Related Topics seminar.** The goal of this lecture is to show how the brownian motion can be derived rigorously from a deterministic system of hard spheres in the limit where the number of particles $N$ tends to infinity, and their diameter simultaneously converges to 0. As suggested by Hilbert in his sixth problem, we will use the linear Boltzmann equation as an intermediate level of description for the dynamics of one tagged particle. We will discuss especially the origin of irreversibility, which is a fundamental feature of both the brownian motion and the Boltzmann equation having no counterpart at the microscopic level.

##### Heat kernel on affine buildings

Let $\mathscr{X}$ be a thick affine building of rank $r+1$. We consider a finite range isotropic random walk on vertices of $\mathscr{X}$. Our main focus is to obtain the optimal global upper and lower bounds for the $n$'th iteration of the transition operator uniform in the region $\text{dist}\big(n^{-1} \delta(O, x), \partial \mathcal{M}\big) \geq Kn^{-1/\eta}$ where $\delta$ is the generalized distance and $\mathcal{M}$ is the convex hull of $\big\{\delta(O, x) : p(O, x) > 0\big\}$.

##### A model for studying double exponential growth

We discuss a model for studying spontaneous phenomena in the 2d Euler equations for incompressible fluid flow. We tie the behaviour of the model to the behavior of the actual Euler equations. (Joint work with A. Tapay.)

##### Asymptotic behaviour of some CMC hypersurfaces of Minkowski space

**Please note special time. ** We study time-like constant mean curvature hypersurfaces of Minkowski space from the point of view of their initial value problems. We focus on initial data close to those generating spherically symmetric expanding solutions, such as anti de Sitter space. Using intuitions from mathematical relativity, we explain how a ``global stability'' statement cannot be true due to the presence of cosmological horizons. Yet using the same intuition we discuss how to easily obtain a spatially localised stability result after a careful geometric reformulation of the problem.

##### Shock Development in Spherical Symmetry

**Please note special time. ** The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but possesses the additional property of having singular initial data due to the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

##### Nonlinear waves on extremal black hole spacetimes

I will discuss the global behaviour of small data solutions to some nonlinear wave equations on certain extremal black hole backgrounds.

##### The entropy production problem in kinetic theory

The Boltzmann equation is the central equation of kinetic theory, and it describes the evolution of the phase space density for a dilute gas. Boltzmann's famous H-theorem says that the entropy is strictly monotone increasing for solutions of this equation unless the solution is in equilibrium. For many years discussion of this result centered on understanding how such irreversibility could arise from reversible Newtonian dynamics, and important questions in this direction remain open. However, more recent work has sought a quantitative version of the H-theorem than can be used to quantify the rate of approach to equilibrium. The first results in this direction were obtained by myself and C. Carvalho, and were further developed and sharpened by Toscani and Villani, and they figure in the work for which Villani won the Fields medal.

##### Vortex filament dynamics

I will describe a a problem in mathematical hydrodynamics, in a setting with a strong analogy to Hamiltonian dynamical systems. The analysis addresses vortex filaments for the three dimensional Euler equations, and a system of model equations for the dynamics of near-parallel filaments. These PDEs can be formulated as a Hamiltonian system, and the talk will describe some aspects of a phase space analysis of solutions, including a theory of periodic and quasi-periodic orbits via a version of KAM theory, a brief description of the relevance of Anderson localization, and a topological principle to count multiplicity of solutions. This is ongoing joint work with C. Garcia (Rome 1 - La Sapienza) and C.-R. Yang (McMaster and the Fields Institute).

##### Ergodic theory for sofic groups and the geometry of model spaces

A large part of classical ergodic theory is concerned with Kolmogorov-Sinai entropy for probability-preserving systems and its consequences. Recent years have seen great progress in generalization some of that theory to actions of more general groups, especially sofic groups. These developments stem from Lewis Bowen's definition of sofic entropy for probability-preserving actions of such groups. This quantity is defined in terms of certain `model spaces' that one can attach to such an action, which consist of all the ways in which that action may be approximated by purely finitary data in a certain way.

##### Infinite energy solutions for the periodic 3D quintic NLS

In this talk we first review recent progress in the study of certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem) which stems from incorporating to the deterministic toolbox, powerful but still classical tools from probability as well. We will explain some of these ideas and describe in more detail joint work with Gigliola Staffilani on the almost sure well-posedness for the periodic 3D quintic nonlinear Schrodinger equation in the supercritical regime; that is, below the critical space $H^1(\mathbb T^3)$. If time permits we will discuss non-deterministic propagation of regularity for NLS in dimensions 1 and 2.

##### Asymptotics of Chebyshev polynomials for general subsets of the real line

Given a compact subset, E in R, the Chebyshev polynomials are the unique minmizers of the sup norm over E among all degree n monic polynomials. I will describe some recent results of Christensen, Simon and inchenko on this classical subject. We settle a 45 year old conjecture of Widom on the large n pointwise asymptotics for the case of finite gap sets. We also extend upper bounds on the norm that Totik and Widom obtained for the finite gap case to positive measure Cantor sets. Our proof in this case is new and even in the finite gap case is simpler and more explicit bounds than the earlier work.