We will discuss new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost all of these sets are covered by regular parabolic Lipschitz graphs, with quantitative control, and that both satisfy parabolic Minkowski bounds depending only on a doubling quantity at a point. Many of these results are new even in the case of the heat equation on $\mathbb{R}^n \times \mathbb{R}$. This is joint work with Max Hallgren and Zilu Ma.

In this talk, I will construct a $p$-adic height pairing for curves with split degenerate stable reduction over a prime $p$ using the higher class field theory of Kato-Saito. This pairing can be shown to coincide with the standard Coleman-Gross height pairing when extended to the semistable reduction case using methods by Besser and Vologodsky. At the end, I will briefly mention how this new method inspires the definition of a height pairing valued in certain Galois groups related to the function field of the original curve.

[pre-talk at 3pm]

I will discuss various problems related to the study of the incompressible Euler equation. The main questions that we will look at have to do with the construction and classification of steady solutions, their stability properties, and the dynamics of nearby unsteady solutions.

Ricci solitons are generalizations of the Einstein manifolds and are self similar solutions to the Ricci flow. In particular, steady Ricci solitons are eternal solutions to the Ricci flow. In this talk, we will discuss the flying wing construction of some Kahler and Riemannian steady Ricci solitons of nonnegative curvature. This is based on joint work with Ronan Conlon and Yi Lai, as well as with Yi Lai and Man Chun Lee.

We review some recent results on the problem of reconstructing a variety from its topology.  This includes some recent work with Fanjun Meng and Lingyao Xie.

I will review recent constructions of oligomorphic tensor categories generalizing Deligne's Rep(S_t). Then, I will lean into the model theoretic part of the question. Specifically, I will explain where there are no continuous families like the original Rep(S_t) and where you should look for n-parameter families, i.e., depending on n free variables. Ultimately, these questions are closely related to classes of structures in model theory.

In this work, we consider an inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture.  Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult.   Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs.

Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement.  To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms with L1 regularization.  We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method.  Numerical experiments illustrate the effectiveness and robustness of the algorithm.

This is a joint work with Samy W. Fung (Colorado School of Mines), Siting Liu (UCR), Levon Nurbekyan (Emory University), and Stanley J. Osher (UCLA).

In 1966, Mark Kac asked the famous question “Can one hear the shape of a drum?”
In his article with this question as the title, he translated it into eigenvalue problems for planar domains.
This question highlighted the relationship between eigenvalues and geometry.
One can then ask how eigenvalues are related to the geometry of the boundary.
In this talk, we consider a special type of eigenvalues, called Steklov eigenvalues, that are closely tied to boundary geometry.
We will introduce Steklov eigenvalues and explain their basic background and applications.
Then we will discuss our recent results on inequalities relating Steklov eigenvalues to the boundary area of compact manifolds.