Analysis Seminar
20192020
Time  Location  Seminar Chair 

Tuesdays at 11am  AP&M 7321  Andrej Zlatoš 
Fall 2019  Winter 2020  Spring 2020 

Fall 2019
Date  Speaker  Title + Abstract 

October 17 (joint with Geometric Analysis) 
Antonio De Rosa
Courant Institute 
Elliptic integrands in analysis
I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by HarrisonPugh and by AlmgrenDavid. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volumeconstrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem. Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli. 
November 7 
Xiaoshan Li
Wuhan University 
Morse inequalities and Kodaira embedding theorems on CR manifolds with group actions
Let $(X, T^{1, 0}X)$ be a compact CR manifold and $(L, h)$ be a Hermitian CR line bundle over $X$. When $X$ is Leviflat and $L$ is positive, Ohsawa and Sibony constructed for every $\kappa\in\mathbb N$ a CR projective embedding of $C^\kappa$smooth of the Leviflat CR manifold. Adachi constructed a counterexample to show that the $C^k$smooth can not be generalized to $C^\infty$smooth. The difficulty comes from the fact that the Kohn Laplacian is not hypoelliptic on Levi flat manifolds. In this talk, we will consider CR manifold $X$ with a transversal CR $G$action where $G$ is a compact Lie group and $G$ can be lifted to a CR line bundle $L$ over $X$. The talk will be divided into two parts. In the first part, we will talk about the Morse inequalities for the Fourier components of KohnRossi cohomology on CR manifolds with transversal CR $S^1$action. By studying the partial Szeg\"o kernel on $(0,q)$forms with values in $L^k$ we obtain the Morse inequalities on $X$ without any Levi form assumption. In the second part, when the CR line bundle $L$ is positive, the Kodaira embedding theorems for CR manifold with $G$action when $G$ is $S^1$, Torus and $\mathbb R$ will be presented. As an application, this will generalize Ohsawa and Sibony's result to $C^\infty$smooth in our setting. 
November 14 
Jiajie Chen
Caltech 
Singularity formation for 2D Boussinesq and 3D Euler equations with boundary and some related 1D models
In this talk, we will discuss recent results on stable selfsimilar singularity formation for the 2D Boussinesq and singularity formation for the 3D Euler equations in the presence of the boundary with $C^{1,\alpha}$ initial data for the velocity field that has finite energy. The blowup mechanism is based on the HouLuo scenario of a potential 3D Euler singularity. We will also discuss some 1D models for the 3D Euler equations that develop stable selfsimilar singularity in finite time. For these models, the regularity of the initial data can be improved to $C_c^{\infty}$. Some of the results are joint work with Thomas Hou and De Huang. 
December 5 in AP&M 6218 
Chris Henderson
University of Arizona 
Wellposedness, blowup, and smoothing for the Landau equation
The Landau equation is a mesoscopic model in plasma physics that describes the evolution in phasespace of the density of colliding particles. Due to the nonlocal, nonlinear terms in the equation, an understanding of the existence, uniqueness, and qualitative behavior of solutions has remained elusive except in some simplified settings (e.g., homogeneous or perturbative). In this talk, I will report on recent progress in the application of ideas of parabolic regularity theory to this kinetic equation. Using these ideas we can, in contrast to previous results requiring boundedness of fourth derivatives of the initial data, construct solutions with low initial regularity (just $L^\infty$) and show they are smooth and bounded for all time as long as the mass and energy densities remain bounded. This is a joint work with S. Snelson and A. Tarfulea. 
December 10 (Tue) at 11am in AP&M 7321 
Christophe Lacave
Universite Grenoble Alpes 
Incompressible Fluids through a Porous Medium
In a perforated domain, the asymptotic behavior of the fluid motion depends on the rate (interhole distance)/(size of the holes). We will present the standard framework and explain how to find the critical rate where "strange terms" appear for the Laplace and NavierStokes equations. Next, we will study Euler equations where the critical rate is totally different than for parabolic equations. These works are in collaboration with V.BonnaillieNoel, M.Hillairet, N.Masmoudi, C.Wang and D.Wu. 
Winter 2020
Date  Speaker  Title + Abstract 

January 7 
Toan Nguyen
Penn State and Princeton 
On Landau damping
The talk presents a quick review on Landau damping for VlasovPoisson system near Penrose stable data, followed by a joint work with D. HanKwan and F. Rousset, where the damping, with screening potential, is proved for data with (essentially) $C^1$ regularity on the whole space. 
January 14 
Peter Polacik
University of Minnesota 
Liouville theorems for superlinear parabolic equations
As in complex analysis, Liouville theorems in PDEs assert that any solution of a specific equation is trivial. The meaning of "trivial" depends on the context. In this lecture, we will discuss Liouville theorems for superlinear parabolic PDEs. An overview, a typical application, and recent results will be presented. 
February 18 
Martin Dindos
University of Edinburgh 
On $p$ellipticity and connections to solvability of elliptic complexvalued PDEs
The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard. In this talk we present a recently discovered structural condition, called $p$ellipticity, which generalizes classical ellipticity. It was codiscovered independently by Carbonaro and Dragicevic on one hand, and Pipher and myself on the other, and plays a fundamental role in many seemingly unrelated aspects of the $L^p$ theory of elliptic complexvalued PDE. So far, $p$ellipticity has proven to be the key condition for: (i) convexity of power functions (Bellman functions) (ii) dimensionfree bilinear embeddings, (iii) $L^p$contractivity and boundedness of semigroups $(P_t^A)_{t>0}$ associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients. During the talk, I will describe my contribution to this development, in particular to (vi). 
March 5 (Thu) at 1pm in AP&M 7218 
Michael Hitrik
UCLA 
Toeplitz operators, asymptotic Bergman projections, and second microlocalization
In the first part of the talk (based on joint work with L.Coburn, J. Sjöstrand, and F. White) we discuss continuity conditions for Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with J. Sjöstrand), we discuss elements of a semiglobal approach to analytic second microlocalization with respect to a hypersurface, in the semiclassical case, based on the study of the heat evolution semigroup for large times. We describe properties of the associated exponentially weighted spaces of holomorphic functions with (h–dependent) plurisubharmonic exponents and construct asymptotic Bergman projections in such spaces. 
March 10 
Stanley Snelson
Florida Tech 
Coercive lower bounds and vacuumfilling in the Boltzmann equation
In this talk, we describe selfgenerating pointwise lower bounds for solutions of the noncutoff Boltzmann equation, which models the evolution of the particle density of a diffuse gas. These lower bounds imply that vacuum regions in the initial data are filled instantaneously, and also lead to key coercivity estimates for the collision operator. As an application, we can remove the assumptions of mass bounded below and entropy bounded above, from the known criteria for smoothness and continuation of solutions. The proof strategy also applies to the Landau equation, and we will compare this (deterministic) proof with our prior (probabilistic) proof of lower bounds for the Landau equation. This talk is based on joint work with Chris Henderson and Andrei Tarfulea. 
March 17 
Ryan Hynd
University of Pennsylvania 
Cancelled 
Spring 2020
(all talks cancelled due to the COVID19 pandemic)
Date  Speaker  Title + Abstract 

March 31 
Javier GomezSerrano
Princeton 
TBA

April 7 
Bjoern Bringmann
UCLA 
TBA

April 14 
Alex Blumenthal
University of Maryland 
TBA

April 21 
Hyunju Kwon
IAS 
TBA

April 28 
James Kelliher
UC Riverside 
TBA

May 5 
Federico Pasqualotto
Princeton University 
TBA

May 12 
Alexander Kiselev
Duke University 
TBA

June 2 
Mohandas Pillai
UC Berkeley 
TBA

Previous years
2018201920172018