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.