Here I shall discuss a few applications related to the heat flow of harmonic map heat flow.

In particular its global existence, uniqueness and regularity into singular spaces; the regularity of maps near sharp interfaces and the behavior of maps near the first time of self-similar blow-ups. The key tool is the monotonicity formulas of Struwe as well as that of  Almgren-Poon type.

A minimizing (1, 2)-cluster is a partition of into one finite volume chamber and two infinite volume chambers that minimizes the interfacial area among all compactly supported variations preserving the volume of the first chamber. For , the “standard lens” is the unique such cluster; its exterior interface is contained in a plane. Starting from , we construct minimizing (1, 2)-clusters that are asymptotic to various prescribed area-minimizing hypercones. This talk is based on joint works with Bronsard, Novack, and Skorobogatova.

Identifying the relationship between the structure of a given rough boundary and the behavior of its elliptic measure has a long history. In the two-phase setting, one typically wishes to deduce geometric information about the boundary from analytic information on the pair of elliptic measures from either side. In the multi-operator setting, the coefficients of the operators from either side may have a discontinuity across the boundary, and thus many of the techniques used in the single-operator setting are not applicable. I will discuss some regularity and structural results, analogous to those in the case of a single operator, which we are able to recover in spite of this. In particular, we will see how to get around the lack of available monotonicity formulas in this setting via improvement of flatness techniques for solutions of a suitable free boundary problem. This is based on joint work with Max Goering, and joint work with Cole Jeznach.

A central theme in modern harmonic analysis and elliptic PDE is the interplay between the analytic properties of boundary value problems and the geometry of the underlying domain. In this talk, I will discuss this principle in the context of the Regularity problem for divergence-form elliptic operators in rough domains.
On the one hand, quantitative geometric assumptions on the boundary, such as uniform rectifiability, imply solvability of the Regularity problem with Sobolev data. On the other hand, solvability itself carries geometric information. I will describe recent results showing that the solvability of the Regularity problem characterizes uniform rectifiability of the boundary, thus establishing an equivalence between analytic and geometric regularity.
I will also discuss a qualitative counterpart of this picture. In this setting, quantitative estimates are replaced by qualitative assumptions, and solvability leads naturally to rectifiability of the boundary. This perspective connects boundary value problems, harmonic measure, and geometric measure theory, and suggests new directions toward free boundary problems where geometry is recovered from analytic information.

In this talk I will give an overview of various recent results concerning quantitative differentiation of Lipschitz and Sobolev functions in geometric contexts. For example, I will present an extension of Dorronsoro theorem to uniformly rectifiable sets and show that all uniformly rectifiable sets may be covered with a surface supporting a Poincaré inequality. I will then apply this theory to problems in unsupervised and supervised statistical learning. These results come from various joint works with J. Azzam, E. Caputo, M. Hyde, M. Mourgoglou, R. Schul and I. Violo.

In the Euclidean setting, David and Semmes conjectured that, in the presence of the ADR condition, boundedness of the Riesz transform implies uniform rectifiability of the underlying measure. We present recent progress on the parabolic analog of this conjecture.

We investigate the asymptotic behavior, as , of solutions to competition-diffusion systems of type

where satisfy the partial segregation condition

in .

For fixed, a solution can be obtained as a minimizer of the functional

on the set of functions in with fixed traces on .

We prove a priori and uniform (in ) Hölder bounds. In the limit, we are led to minimize the energy

over all partially segregated states

in

satisfying the given partially segregated boundary conditions above. We prove regularity of the free boundary up to a low-dimensional singular set.

Since elliptic PDE theory can be seen as a steady-state version of parabolic PDE theory, if a parabolic estimate holds, then by eliminating the time parameter, one can obtain an underlying elliptic statement. Producing a parabolic statement from an elliptic statement, on the other hand, is not as straightforward. In this talk, we will discuss how a high-dimensional limiting technique can be used to prove theorems about solutions to the fractional heat equation from their elliptic counterparts. This is a joint work with Blair Davey.

In the study of the Bernstein problem, Bombieri-De Giorgi-Giusti established the existence of minimal foliations around the Simons cone. Later, Hardt-Simon identified a remarkable condition, the strictly minimizing condition, which determines the decay rate of smooth minimal hypersurfaces in this foliation as they approach the minimal cone, i.e. if it is a slow decay or fast decay. This condition proves highly useful in numerous other problems, such as the construction of entire minimal graphs. In this talk, I will first provide a review of this slow decay vs. fast decay issue, and then discuss similar phenomena in several other problems, including the Allen-Cahn equation and the Alt-Caffarelli one-phase free boundary problem.