Calendar
Posted January 12, 2025
Last modified April 29, 2025
Zi Li Lim, UCLA
Rational function progressions
Szemeredi proved that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions. Subsequently, Szemeredi's theorem was generalized to the polynomial and multidimensional settings. We will discuss finding the progressions involving rational functions via Fourier analysis and algebraic geometry.
Posted May 2, 2025
Last modified May 9, 2025
Felipe Ramirez, Wesleyan University
Higher dimensional and moving target versions of the Duffin--Schaeffer conjecture
The Duffin--Schaeffer conjecture (1941) was one of the most pursued problems in metric Diophantine approximation, until it was proved by Koukoulopoulos and Maynard in 2020. Roughly speaking, it gives a precise criterion to determine whether almost all or almost no real numbers are approximable by rationals at a given rate. In this talk I will introduce the problem and its context, and I will discuss higher dimensional and inhomogeneous versions of it, including some problems that are still open. Parts of the talk are based on joint work with Manuel Hauke.
Posted September 10, 2025
Last modified October 14, 2025
Bruno Poggi, University of Pittsburgh
The Dirichlet problem as the boundary of the Poisson problem
We review certain classical quantitative estimates (known as non-tangential maximal function estimates) for the solutions to the Dirichlet boundary value problem for the Laplace equation in a smooth domain in Euclidean space, when the boundary data lies in an $L^p$ space, $p>1$. A natural question that arises is: what might an analogous estimate for the inhomogeneous Poisson problem look like? We will answer this question precisely, and in so doing, we will unravel deep and new connections between the solvability of the (homogeneous) Dirichlet problem for the Laplace equation with data in $L^p$ and the solvability of the (inhomogeneous) Poisson problem for the Laplace equation with data in certain Carleson spaces. We employ this theory to solve a 20-year-old problem in the area, to give new characterizations and a new local T1-type theorem for the solvability of the Dirichlet problem under consideration. Some of the new results are the product of joint works with Mihalis Mourgoglou and Xavier Tolsa.
Event contact: Phuc C. Nguyen
Posted March 3, 2026
Last modified March 9, 2026
Shuang Guan, Tufts University
The HRT Conjecture for a Symmetric (3,2) Configuration
The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved. In this talk, I will discuss the HRT conjecture for a specific symmetric configuration of five points in the time-frequency plane, known as the (3,2) configuration. We prove that for this specific setting, the Gabor system is linearly independent whenever the parameters satisfy certain rationality conditions (specifically, when one parameter is irrational and the other is rational). This result partially resolves the remaining open cases for such configurations. I will outline the proof methods, which involve an interplay of harmonic analysis and ergodic theory. This is joint work with Kasso A. Okoudjou.
Posted March 1, 2026
Last modified March 26, 2026
Simon Bortz, University of Alabama
Parabolic Quantitative Rectifiability, Singular Integrals, and PDEs
I will discuss the origins of quantitative rectifiability, starting with the Littlewood–Paley g-function and the Fefferman–Stein characterization of BMO via Poisson extensions. From this point of view, I will describe some of the motivations behind the David–Semmes characterization of uniform rectifiability in terms of Jones’ $L^2$ beta numbers. I will then discuss my work establishing parabolic analogues of some of the equivalences proved by David and Semmes in the elliptic setting, as well as related work by others. I will conclude with recent work connecting this theory to the Dirichlet problem for the heat equation and to quantitative properties of caloric functions.
Posted March 9, 2026
Last modified April 6, 2026
Alexander Burgin, Georgia Tech
Integer Cantor sets: Harmonic-analytic properties & arithmetic applications.
Integer Cantor sets, which consist of a set of integers in a fixed base and a fixed set of digits, have many interesting properties, including uniform distribution, metric pair correlation, and mean ergodic theorems. In particular, their Fourier transform factorizes. I’ll begin with a motivation from ergodic theory, and proceed to discuss some recent results of myself, Fragkos, Lacey, Mena, and Reguera. If time permits, I will discuss some arithmetic applications of these estimates.