# Analysis Seminar

## 2023-2024

Time | Location | Organizers |
---|---|---|

Tuesdays at 11am | APM 7321 or Zoom (contact organizers for meeting ID) | Ioan Bejenaru and Andrej Zlatoš |

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### 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 $L^p$ 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 2024-25 |

### 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 dbar-Neumann problem. The latter is joint work with Samuele Mongodi (Univ. Milano-Bicocca, 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 80s-90’s that solutions can exhibit bubbling along a well-chosen sequence of times — the solution decouples into a superposition of well-separated 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 multi-bubbles 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 |
Song-Ying Li
UC Irvine |
Sup-norm 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 sup-norm estimates for the Cauchy-Riemann 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 so-called 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 d-bar-Neumann 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 multi-type''. The finiteness of the tower multi-type 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 multi-type, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's celebrated potential-theoretic ``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 one-sided two phase Bernoulli free boundary problem
We study a two-phase 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 one-phase part of the free boundary and a two-phase condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity 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 Hele-Shaw 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 |
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 reverse-time diffusion processes, the method computes concentrated invariant measures mesh-free up to dimension 16 at a linear complexity rate based on solving a principal eigenvalue problem of non-self-adjoint advection-diffusion 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 self-similarity. |

May 21 |
Anuj Kumar
UC Berkeley |
Sharp nonuniqueness of the transport equation with Sobolev vector fields
We construct nonunique solutions of the transport equation in the class $L^\infty$ in time and $L^r$ in space, for divergence free Sobolev vector fields from $W^{1, p}$. We achieve this by introducing two novel ideas: (1) in the construction, we interweave scaled copies of the vector field itself, and (2) asynchronous translation of cubes, which makes the construction heterogeneous in space. These new ideas allow us to prove nonuniqueness in the range of exponents going beyond what is available using the method of convex integration, and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis ’21. |

June 4 at 2pm in APM 6402 |
Zhenghui Huo
Duke Kunshan University |
Weighted estimates of the Bergman projection and some applications.
In harmonic analysis, the Muckenhoupt $A_p$ condition characterizes weighted spaces on which classical operators are bounded. An analogue $B_p$ condition for the Bergman projection on the unit ball was given by Bekolle and Bonami. As the development of the dyadic harmonic analysis techniques, people have made progress on weighted norm estimates of the Bergman projection for various settings. In this talk, I will discuss some of these results and outline the main ideas behind the proof. I will also mention the application of these results in analyzing the $L^p$ boundedness of the projection. This talk is based on joint work with Nathan Wagner and Brett Wick. |

June 6 (Thu) at 1pm in APM 7218 |
Siao-Hao Guo
National Taiwan University |
Level set flow and the set of singular points
In this talk we will introduce the level set flow. Then we will talk about the relation between the rate of curvature blow-up near a singularity of the flow and the distribution of surrounding singular points. |

### Previous years

2022-20232021-2022

2019-2020

2018-2019

2017-2018