# Analysis Seminar

## 2019-2020

Time Location Seminar Chair
Tuesdays at 11am     (from Winter 2020) 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 Harrison-Pugh and by Almgren-David. 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 volume-constrained 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 Levi-flat and $L$ is positive, Ohsawa and Sibony constructed for every $\kappa\in\mathbb N$ a CR projective embedding of $C^\kappa$-smooth of the Levi-flat 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 Kohn-Rossi 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 self-similar 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 Hou-Luo scenario of a potential 3D Euler singularity. We will also discuss some 1D models for the 3D Euler equations that develop stable self-similar 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
Well-posedness, blow-up, and smoothing for the Landau equation
The Landau equation is a mesoscopic model in plasma physics that describes the evolution in phase-space of the density of colliding particles. Due to the non-local, non-linear 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 (inter-hole 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 Navier-Stokes 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.Bonnaillie-Noel, 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 Vlasov-Poisson system near Penrose stable data, followed by a joint work with D. Han-Kwan 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 complex-valued 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 co-discovered 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 complex-valued PDE. So far, $p$-ellipticity has proven to be the key condition for: (i) convexity of power functions (Bellman functions) (ii) dimension-free 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 vacuum-filling in the Boltzmann equation
In this talk, we describe self-generating pointwise lower bounds for solutions of the non-cutoff 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 are likely cancelled due to the COVID-19 epidemic)

Date Speaker Title + Abstract
March 31 Javier Gomez-Serrano
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

2018-2019
2017-2018