Abstract: A theorem of Besicovitch from the 30s states that a planar set with finite length and “many” projections of positive measure has a rectifiable piece. How big is this piece, relative to the measure of the projections? In general, quantifying Besicovitch’s theorem remains an open problem, but I will discuss a recent partial result: n-regular sets in Rd with “plenty of big projections”, in the sense of David and Semmes, contain big pieces of Lipschitz graphs.

Abstract: In recent years P. Gressman, in the context of the L^p-improving problem for Radon averages, has introduced two types of weighted surface measures. One is an affine-invariant surface measure (of "best-possible" type) for surfaces of arbitrary codimension, obtained by a clever construction related to Geometric Invariant Theory (GIT). The other arises via a non-degeneracy condition that enables an inflation method devised to prove L^p-improving inequalities (this is the antecedent of the work that P. Gressman presented in his talk on 27/1/2021).

We pose the question of what the relationship between the two weights is and provide some partial answers. It is a matter of a simple calculation to verify that in codimension 1 the weights are comparable, but the situation in higher codimensions is much less clear - sometimes the comparability fails. Using GIT techniques, we are able to show the weights continue to be comparable in codimension 2 (in even ambient dimension). (Joint work with S. Dendrinos and A. Mustata)

Abstract: A natural 3-dimensional analogue of Bourgain’s circular maximal function theorem in the plane is the study of the sharp L^p bounds in R^3 for the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). In this talk, we present a sharp result, which establishes that L^p bounds hold if and only if p>3. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

Abstract: We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions. This is joint work with Terence Tao.

Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.

Abstract: Various problems in harmonic analysis are intimately connected with studying solutions to additive equations over subsets of curves and surfaces. The latter is amenable to techniques from incidence geometry since we can count such solutions by interpreting them as incidences between points and curves/surfaces. In this talk, we study additive energies of arbitrary subsets of parabolas/convex curves, and their connections to a problem of Bourgain and Demeter regarding a diameter-free version of the quadratic Vinogradov mean value theorem. We also mention some new results associated with additive energies on higher dimensional surfaces which are related to restriction type problems on spheres.

Abstract: We show that, when N is any square-free integer, the size of the smallest Kakeya set in (ℤ/Nℤ)^n is at least C_{eps,n}*N^{n-eps} for any eps>0 -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the p-rank of the incidence matrix of points and hyperplanes over (ℤ/p^kℤ)^n. Joint work with Manik Dhar

Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.

Abstract: The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions.

In the talk, we will build up some intuition about waves to motivate the conjecture, and then discuss some of the obstacles and some ideas from the proof.

Abstract: I will discuss a discretized version of the Elekes-Ronyai theorem from additive combinatorics, which is closely related to the sum-product problem. The Elekes-Ronyai theorem has recently had applications to combinatorial geometry, including variants of the Erdos distinct distances problem. The discretized version of the Elekes-Ronyai theorem has similar applications, and in particular I will discuss some new results on a pinned version of the Falconer distance problem. This is joint work with Orit Raz.

Abstract: Fourier restriction, Radon-like operators, and decoupling theory are three active areas of harmonic analysis which involve submanifolds of Euclidean space in a fundamental way. In each case, the mapping properties of the objects of study depend in a fundamental way on the "non-flatness" of the submanifold, but with the exception of certain extreme cases (primarily curves and hypersurfaces), it is not clear exactly how to quantify the geometry in an analytically meaningful way. In this talk, I will discuss a series of recent results which shed light on this situation using tools from an unusually broad range of mathematical sources.

Abstract: In his recent paper Josh Zahl proves (among other things) a new single distance bound n^{3/2} for a set of n points in a 3-space over a field F, where -1 is not a square. In his considerations he implicitly uses the concept of a line complex, which has many interesting properties. I will present his result in this light and extend it to a weaker bound n^{1.6} over F, where -1 is a square.

Abstract: We develop a theory for oscillatory integrals which can be applied in a variety of settings, especially settings where scale-invariant bounds do not hold in the generality we are accustomed to.