The dimension of harmonic measure in is the maximal minimum Hausdorff dimension of a subset of a boundary of a bounded domain through which the trace of Brownian motion first exits the domain almost surely. From theorems of Makarov, Jones, Wolff, and Bourgain in the 1980s, it is known that and for all .

In the early 1990s, Bishop conjectured that and for all , but the conjecture is wide open. I will discuss ideas behind the state-of-the-art upper bounds and for all , including building covers with disjoint cubes in general position and a flexible scheme to approximate polar integration against Ahlfors regular measures.

I will also present several challenges on the dimension of harmonic measure that could plausibly be solved before Bishop’s conjecture. This talk is based on joint work with Michael Albert (University of Connecticut, Okinawa Institute of Science and Technology) and Alyssa Genschaw (Milwaukee School of Engineering).

This is joint work with Polina Perstneva, to be written (so, not yet secured but probably interesting anyway). The initial question concerned the construction, on domains of the plane bounded by general snowflakes, of elliptic operators with a given harmonic measure on the boundary. For this we like to construct the level sets of the Green function first, and the desired ellipticity of the constructed operator translates into a long distance regularity of the distance between the orthogonal gradient lines. A regulation tool is needed, which translates into the following control problem on the square. Can we find two mutually orthogonal sets of curves in the unit square (green and red), so that the green curves connect points on the left to on the right, and the red ones go from on the bottom to on the top, where is any given small enough compact perturbation of the identity in ?

The free boundary in the classical obstacle problem can have cusps with unbounded curvature in dimensions three and higher. I will discuss how its mean curvature nevertheless admits a universal upper bound in every dimension. The estimate follows from an optimal -semiconvexity bound for the solution. Its proof is something of a guided tour through the modern free-boundary toolbox: frequency formulas, improvement of semiconvexity, (log-)epiperimetric inequalities, spectral gaps, and various linearizations all make an appearance. I will focus on how these ingredients fit together as well as the new ideas behind the proof.

Joint work with Giacomo Colombo.

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 will discuss the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast to the melting version of the problem, the problem exhibits non-uniqueness, infinite speed of propagation and nucleation of ice. We will discuss the geometry of the water-ice interface (free boundary) and the structure of singularities, with the notion of maximal solutions.

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 behavior of maps near the first time self-similar blow-ups and the uniqueness of tangent flows; and properties of Rank 0 and Rank 1 sets of maps. The key tool is the monotonicity formulas of Struwe as well as that of Almgren-Poon type.

I will talk about the behavior of tracers (solutions to a parabolic PDE with a turbulent drift) and related geometric problems. This is joint work with Charles Fefferman.

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.

We study energy minimizers for quasilinear one-phase free boundary problems. We show that free boundaries are regular outside a singular set of codimension at least . This talk is based on joint work with A. De Rosa and H. Yu.

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.

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.

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

where satisfy the partial segregation condition

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:

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

Joint works with Nicola Soave.

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.

The abstract as a PDF

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.

In this talk, we discuss the rigidity of viscosity solutions to the Bernoulli problem whose free boundary is a graph. In low dimensions, we show that such free boundaries must be hyperplanes. In high dimensions, we construct a solution with nontrivial graphical free boundaries.

This talk is based on joint works with Max Engelstein (Minnesota), Xavier Fernández-Real (EPFL) and Kelei Wang (Wuhan).