Analysis Seminar
20232024
Time  Location  Organizers 

Tuesdays at 11am  APM 7321 or Zoom (contact organizers for meeting ID)  Ioan Bejenaru and Andrej Zlatoš 
Fall 2023
Date  Speaker  Title + Abstract 

October 17 on Zoom 
Yannick Sire
Johns Hopkins University 
Spectral estimates for Schrödinger operators on manifolds
I will report on recent results stemming from the analysis of Schrödinger operators on manifolds. I will first describe results dealing with isoperimetric inequalities and optimal (aka extremal) metrics on closed manifolds. These issues have been instrumental in the study of the spectrum of several classical operators, and are motivated by the understanding of the behaviour of the spectrum under changes of metrics. Then, motivated by conjectures of Yau on measures of nodal sets (but which are actually related to the first part of the talk), I will describe how eigenfunctions are concentrating in terms of Lp norms (with an explicit dependence on the eigenvalues). My goal is to emphasize on the case of Schrödinger operators with rough potentials. I will also state several open problems. 
November 7 
Jack Xin
UC Irvine 
Postponed to May 14 
November 14 
Vladimir Sverak
University of Minnesota 
Postponed to 202425 
Winter 2024
Date  Speaker  Title + Abstract 

January 10 (Wed) at 3pm in APM 7218 
Gian Maria Dall’Ara
Indam at Scuola Normale Superiore 
An uncertainty principle for the dbar operator
I will present a rather elementary inequality and discuss its application to dbar equations with weights on the whole complex Euclidean space and to subelliptic estimates for the dbarNeumann problem. The latter is joint work with Samuele Mongodi (Univ. MilanoBicocca, Italy). 
January 10 (Wed) at 4pm in APM 7218 
Weixia Zhu
University of Vienna 
Deformation of CR structures and Spectral Stability of the Kohn Laplacian
The interplay between deformation of complex structures and stability of spectrum for the complex Laplacian on compact complex manifolds was studied extensively by Kodaira and Spencer in the 1950s. In this talk, we will discuss analogous problems for complex manifolds with boundaries and for compact CR manifolds. This talk is based on joint work with Howard Jacobowitz and Siqi Fu. 
January 23 at 4:00pm in APM 6402 
Andrew Lawrie
MIT 
Continuous bubbling for the harmonic map heat flow
I will discuss joint work with Jacek Jendrej and Wilhelm Schlag about the two dimensional harmonic map heat flow for maps taking values in the sphere. It has been known since the 80s90’s that solutions can exhibit bubbling along a wellchosen sequence of times — the solution decouples into a superposition of wellseparated harmonic maps and a body map accounting for the rest of the energy. We prove that every sequence of times contains a subsequence along which such bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multibubbles in continuous time. The proof is partly motivated by the classical theory of dynamical systems and uses the notion of “minimal collision energy” developed in joint work with Jendrej on the soliton resolution conjecture for nonlinear waves. 
February 16 (Fri) at 3pm in APM 7218 
SongYing Li
UC Irvine 
Supnorm Estimates for $\overline{\partial}$ and Corona Problems
In this talk, we will present the development of Corona problem in sereval complex variables and discuss its relation to the solution of the supnorm estimates for the CauchyRiemann equations. It includes the Berndtsson conjecture and its application to Corona problem. As well as the application of the H\"ormander weighted $L^2$estimates for $\overline{\partial}$ to the corona problem. 
February 16 (Fri) at 4pm in APM 7218 
Min Ru
University of Houston 
Recent developments in the theory of holomorphic curves
In this talk, I will discuss some recent developments and techniques in the study of the theory of holomorphic curves (Nevanlinna theory). In particular I will discuss the recent techniques of the socalled G.C.D. method as the applications of my recent work with Paul Vojta. 
February 27 
Dmitri Zaitsev
Trinity College Dublin 
Global regularity in the dbarNeumann problem and finite type conditions
The celebrated result of Catlin on global regularity of the $\bar\partial$Neumann operator for pseudoconvex domains of finite type links local algebraic and analytic geometric invariants through potential theory with estimates for $\bar\partial$equation. Yet despite their importance, there seems to be a major lack of understanding of Catlin's techniques, resulting in a notable absence of an alternative proof, exposition or simplification. The goal of my talk will be to present an alternative proof based on a new notion of a ``tower multitype''. The finiteness of the tower multitype is an intrinsic geometric condition that is more general than the finiteness of the regular type, which in turn is more general than the finite type. Under that condition, we obtain a generalized stratification of the boundary into countably many level sets of the tower multitype, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's celebrated potentialtheoretic ``Property (P)'', which, in turn, is known to imply global regularity via compactness estimate. Notable applications of global regularity include Condition R by Bell and Ligocka and its applications to boundary smoothness of proper holomorphic maps generalizing a celebrated theorem by Fefferman. 
February 27 at 3pm in APM 6218 
John D'Angelo
UIUC 
Some old work of Kohn and some of my contributions
I will discuss in detail some work of Joe Kohn involving subelliptic estimates. I hope to provide an understandable account of some of the technical matters from his 1979 Acta paper, and I will discuss some of my own work on points of finite type. Although there is no new theorem to present, I will provide several new approaches to these ideas. To prevent the talk from being too technical, I will also include several elementary interludes that can be understood by graduate students. 
March 7 (Thu) at 4:00pm in APM 6402 
Gunther Uhlmann
University of Washington 
Journey to the Center of the Earth
We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It also has several applications in optics and medical imaging among others. The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will survey some of the known results about this problem. No previous knowledge of differential geometry will be assumed. 
Spring 2024
Date  Speaker  Title + Abstract 

April 16 
Lorenzo Ferreri
Scuola Normale Superiore di Pisa 
A onesided two phase Bernoulli free boundary problem
We study a twophase free boundary problem in which the two phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the onephase part of the free boundary and a twophase condition on the collapsed part of the free boundary. For this twomembrane type problem, we prove an epsilonregularity theorem in all dimensions. Then, in dimension two, we study the fine structure of the free boundary, showing that the branching points can only occur in finite number. This is a joint work with Luca Spolaor and Bozhidar Velichkov. 
May 7 
Inwon Kim
UCLA 
A model on tumor growth with nutrient
We discuss a model that describes tumor growth with nutrient, where the tumor density satisfies an “incompressibility” condition. The model, which can be formulated as a HeleShaw type free boundary problem, generates evolution of tumor patches, where its density is uniform. We will discuss interesting features on the dynamics of tumor patch evolution, some established results, and some open questions. 
May 14 on Zoom 
Jack Xin
UC Irvine 
Computing Entropy Production Rates and Chemotaxis Dynamics in High Dimensions
by Stochastic Interacting Particle Methods
We study stochastic interacting particle methods with and without field coupling for high dimensional concentration and singularity formation phenomena. In case of entropy production of reversetime diffusion processes, the method computes concentrated invariant measures meshfree up to dimension 16 at a linear complexity rate based on solving a principal eigenvalue problem of nonselfadjoint advectiondiffusion operators. In case of fully parabolic chemotaxis nonlinear dynamics in 3D, the method captures critical mass for finite time singularity formation and blowup time at low costs through a smoother field without relying on selfsimilarity. 
May 21 
Anuj Kumar
UC Berkeley 
TBA

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