In this talk I will describe a recent characterization of uniform rectifiability in terms of a variant of the so called area function (or g function) for harmonic functions and an analogous result involving solutions of elliptic PDE's. As a byproduct, we obtain a new characterization of uniform rectifiability in terms of harmonic measure or elliptic measure. Some of these results are joint work with Azzam, Garnett and Mourgoglou

Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou, Yannis Parissis and Gennady Uraltsev

We will use the notion of rigidity, group actions and Fourier analysis to study the problem of the existence of congruent copies of finite point configurations of a given type inside sets of a given Hausdorff dimension

The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a square sum of quantities called beta-numbers that measure how non-flat the set is at each scale and location. Conversely, given such a curve, the square sum of its beta-numbers is controlled by the total length of the curve, giving us quantitative information about how non-flat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional surfaces. This is joint work with Raanan Schul.

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$,  and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.

The homogeneous Sobolev spaces $\dot{W}^{s,n/s}(\mathbb{R}^n)$ all fail to embed into $L^{\infty}$ when $0<s<n$, but only barely so. In dimension $n \geq 2$, Bourgain and Brezis found a rather useful remedy for this failure when $s = 1$, by considering how well a general $\dot{W}^{1,n}$ function can be approximated by an $L^{\infty}$ function on $\mathbb{R}^n$. In this talk we give an extension of their results to $\dot{W}^{s,n/s}$ for all $0 < s < n$. This is joint work with Pierre Bousquet, Emmanuel Russ and Yi Wang.

In this talk, we will discuss how decoupling inequalities allow for the establishment of well-posedness results for the nonlinear Schrodinger equation in the case when data is defined on a two-dimensional irrational torus.