# Seminars & Events for Analysis Seminar

##### The characteristic gluing problem for the wave equation and applications

We will first describe the characteristic gluing problem for the wave equation on a general four-dimensional Lorentzian manifold. We will show that the only obstruction to such gluing constructions is in fact the existence of certain ``conservation laws'' on null hypersurfaces and we will then obtain necessary and sufficient conditions for the existence of such conservation laws. Our method relies on a novel elliptic structure associated to a foliation with 2-spheres of a null hypersurface. We will finally present some applications to black hole spacetimes.

##### Random weighted Sobolev inequalities

We extend a randomisation method, introduced by Burq-Lebeau on compact manifolds, to the case of the harmonic oscillator. We construct measures, under concentration of measure type assumptions, on the support of which we prove optimal weighted Sobolev estimates on R^d. As an application we can prove almost sure global well posedness results for the nonlinear Schrödinger equation with harmonic potential. This is a joint work with Aurélien Poiret and Didier Robert.

##### Formation of shocks for quasilinear wave equations

For any nonzero real constant $c_0$, we exhibit a family of smooth initial data for $$\big(-1 + c_0 (\partial_t \varphi)^2\big)\partial_t^2 \varphi + \triangle \varphi = 0 $$ and show that shocks form in the future. No symmetry condition is assumed. The work combines ideas from fluid mechanics, e.g. shock formation for Euler equations, and from general relativity, e.g. nonlinear stability of Minkowski spacetime and formation of trapped surfaces. This is joint with S. Miao.

##### Well-posedness for Euler 2D in non-smooth domains

The well-posedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Stability of the neutral state for the 2 fluid Euler-Maxwell system

This is a joint work with Y. Guo and A. Ionescu. We consider the 2 fluid Euler-Maxwell problem in the whole 3d space. We show that small and smooth irrotational initial perturbations lead to solutions which remain globally smooth and scatter back to equilibrium. This shows a stabilization effect due to the electromagnetic field since in the absence of electromagnetic field, a classical result of Sideris shows that formation of shock is expected for compressible fluids. The proof consists of a combination of energy estimates to provide a first rough control and nonlinear stationary phase estimates to obtain decay of the solution. However the presence of electron accoustic wave introduces new degeneracies in the phase functions.

##### Higher Singular Integrals

There is a very natural way to extend Calder\'{o}n's calculations which generated his commutators and the Cauchy integral on Lipschitz curves, to include operators of multiplication with functions of arbitrary polynomial growth. The plan of the talk is to describe some examples of such calculations, and to focus on a particular case that goes beyond the classical Calder\'{o}n program."

##### Stable small-data shock formation for wave equations in 3D

PLEASE NOTE SPECIAL DAY AND TIME. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. I will present some preliminary results obtained in collaboration with G. Holzegel, S. Klainerman, and W. Wong. Our main result is a proof of stable shock formation in solutions to a class of nonlinear wave equations in three spatial dimensions.

##### Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

We study time-like hypersurfaces with vanishing mean curvature in the $(3+1)$ dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain class of symmetry the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid. This is joint work with Roland Donninger, Joachim Krieger and Willie Wong.

##### Global solutions and asymptotic behavior for two dimensional gravity water waves

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description of the solution which shows that modified scattering holds. The main tools used in the proof are, on the one hand, a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. On the other hand, we prove uniform bounds interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical Lagrangian distributions.

##### Higher-order analogues of the exterior derivative complex

I will begin by presenting earlier joint work with E. M. Stein concerning div-curl type inequalities for the exterior derivative complex and its adjoint in Euclidean space R^n. I will then offer various, higher-order generalizations of the notion of exterior derivative that stem from recent work of Bourgain-Brezis and Van Schaftingen, and discuss related div-curl type estimates for such operators. Part of this work is joint with A. Raich.

##### Global well-posedness and scattering for the defocusing, mass critical generalized KdV problem

The mass - critical gKdV has many features in common with the mass critical NLS problem. In particular, we show that scattering for the quintic problem implies a concentration compactness result for the gKdV problem. We then define an interaction Morawetz estimate that implies scattering.

##### Regularity theory for a class of fully nonlinear integral operators

We consider a class of non-linear integral variational problems involved in nonlocal image and signal processing. We show the existence of global in time classical solutions for those problems. The method is based on the De Giorgi method applied to nonlocal operators. It is an extension of a similar result we first obtained for the the so-called Surface Geostrophic Equation.

##### Global regularity for 2d water waves

We consider the water waves system for the evolution of a perfect irrotational fluid with a free boundary, in 2 spatial dimensions, under the influence of gravity. We prove the existence of global solutions for suitably small and regular initial data. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the 3 dimensional case. This is joint work with A. Ionescu.

##### Linear Stability of the Schwarzschild Solution under gravitational perturbations

I will talk about recent work joint with M. Dafermos and I. Rodnianski establishing the linear stability of the Schwarzschild solution. Key to the proof is a novel quantity which decouples from the system of gravitational perturbations and satisfies a wave equation, for which decay estimates can be proven. I will also connect the result to the non-linear stability problem.

##### Local well-posedness for the equation of minimal hypersurface in Minkowski space

A timelike minimal hypersurface in Minkowski space satisfies a quasilinear wave equation. I will explain how the minimal hypersurface equation exhibits a null structure and how to utilize the null structure in order to lower the regularity requirements on the initial data for the Cauchy problem.

##### What does it mean to solve KPZ?

The last few years have seen significant progress in understanding the KPZ equation and its universality class, from the meaning of the equation, to a big picture conjecture of asymptotic fluctuations, to several exact formulas. We will try to give a survey of some of this progress, and use recent results on flat exclusion to illustrate some of the exact solvability.

##### Scalar curvature rigidity and flexibility phenomena in asymptotically flat spaces

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. A well-known corollary of the positive mass theorem by Schoen-Yau is that if an asymptotically flat manifold (of non-negative scalar curvature) is exactly flat outside of a compact set, then it has to be globally flat: in other terms any such metric can never be localized inside a compact set

##### Unique continuation from infinity for linear waves

**This is an additional Analysis seminar. **I present various uniqueness results from null infinity, for linear waves on asymptotically flat spacetimes. Assuming vanishing of a solution to infinite order on suitable parts of future and past null infinity, we derive that the solution must vanish in a domain in the interior. I will elaborate on the role of the background geometry in this problem. In particular, it turns out that for spacetimes with positive mass the required assumptions are much weaker than those required for Minkowski spacetime. The results are nearly optimal in many respects. This work is partly motivated by questions in general relativity, and was obtained jointly with Spyros Alexakis and Arick Shao.

##### On some problems in pointwise ergodic theory

**Please note special day (Wednesday).**

##### The discrete spherical maximal function and Waring's problem

**Please note special day (Friday). **We will start by motivating some problems in discrete harmonic analysis by discussing their Euclidean counterparts. Then I will discuss Stein's spherical maximal function, Magyar's discrete version and the ideas behind Magyar-Stein-Wainger's theorem (proving L^p boundedness). We will pay particular attention to the synthesis of ideas from harmonic analysis and analytic number theory. I will then discuss higher degree versions and lacunary versions. I will conclude with applications to ergodic theory and combinatorics. If time permits, we will discuss the connection to restriction theorems.