Calendar
Calendar
Posted February 23, 2025
Last modified February 24, 2025
Hamed Musavi, King's College London
An overview on the recent progress on quantitative Szemeredi Theorems
In this talk, we will start with introducing the classical (qualitative) Ramsey-type Theorems in Additive Combinatorics such as Roth, Sarkozy and Szemeredi Theorems. Then we propose the quantitative problems and a motivation behind their importance. Next, we mention a few recent results on these problems. Finally if time permits, we will talk about ideas in the proofs. This is a joint work with Ben Krause, Terence Tao, and Joni Teravainen.
Posted March 8, 2025
Last modified March 9, 2025
Tomoyuki Kakehi, University of Tsukuba
Snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation
In this talk, we deal with snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation. For simplicity, let us consider the wave equation $\partial_t^2 u - \Delta u =0$ on $\mathbb{R}^n$ with the condition $u|_{t=t_1} =f_1, \cdots, u|_{t=t_m} =f_m$. It is natural to ask when the above equation has a unique solution. We call the above problem the snapshot problem for the wave equation, and call the set of $m$ functions $\{ f_1, \cdots, f_m \}$ the snapshot data. Roughly speaking, one of our main results is as follows. {\bf Theorem.} We assume that $m=3$ and $(t_3-t_1)/(t_2 -t_1)$ is irrational and not a Liouville number. In addition, we assume a certain compatibility condition on the snapshot data $\{ f_1, f_2, f_3 \}$. Then the snapshot problem for the wave equation has a unique solution. We also consider a similar snapshot problem for the Euler-Poisson-Darboux equation. This is a joint work with Jens Christensen, Fulton Gonzalez, and Jue Wang.
Posted March 9, 2025
Last modified April 9, 2025
Tomoyuki Kakehi, University of Tsukuba
Inversion formulas for Radon transforms and mean value operators on the sphere
This talk consists of two parts. In the first part, we explain the Radon transfrom associated with a double fibration briefly and then we introduce several inversion formulas. In the second part, we deal with the mean value operator $M^r$ on the sphere. Here we define $M^r: C^{\infty} (\mathbb{S}^n) \to C^{\infty} (\mathbb{S}^n)$ by $$ M^r f (x) = \frac{1}{\mathrm{Vol} (S_r (x))} \int_{y \in S_r (x)} f(y) d\mu(y), \qquad f \in C^{\infty} (\mathbb{S}^n), $$ where $S_r (x)$ is the geodesic sphere with radius $r$ and center $x$ and $d\mu$ is the measure on $S_r (x)$ induced from the canonical measure on $\mathbb{S}^n$. We will give conditions on $r$ for $M^r$ being injective or surjective. For example, in the case $n=3$, $M^r$ is injective but not surjective if and only if $r/\pi$ is a Liouville number. We will also give some related results on Gegenbauer polynomials. This is a joint work with J. Christensen, F. Gonzalez, and J. Wang.
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