Posted January 24, 2024
Last modified March 13, 2024
Alan Chang, Washington University in St. Louis
Venetian blinds, digital sundials, and efficient coverings
Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.
Posted October 15, 2024
Last modified October 22, 2024
Nathan Mehlhop, Louisiana State University
Ergodic averaging operators
Certain quantitative estimates such as oscillation inequalities are often used in the study of pointwise convergence problems. Here, we study these for discrete ergodic averaging operators and discrete singular integrals along polynomial orbits in multidimensional subsets of integers or primes. Because of its relevance to multiparameter averaging operators, we also consider the vector-valued setting. Several tools including the Hardy-Littlewood circle method, Weyl's inequality, the Ionescu-Wainger multiplier theorem, the Magyar-Stein-Wainger sampling principle, the Marcinciewicz-Zygmund inequality, and others, are important in this field. The talk will introduce the problem and many of these ideas, and then give some outline of how the various estimates can be put together to give the conclusion.
Posted October 12, 2024
Last modified October 30, 2024
Vishwa Dewage, Clemson University
The Laplacian of an operator and applications to Toeplitz operators
Werner's quantum harmonic analysis (QHA) provides a set of tools that are applicable in many areas of analysis, including operator theory. As noted by Fulsche, QHA is particularly suitable to study Toeplitz operators on the Fock space. We explore the Laplacian of an operator and a heat equation for operators on the Fock space using QHA. Then we discuss some applications to Toeplitz operators. This talk is based on joint work with Mishko Mitkovski.
Posted October 12, 2024
Last modified October 22, 2024
Robert Fulsche, University of Hannover, Germany
Harmonic analysis on phase space and operator theory
In his paper \emph{Quantum harmonic analysis on phase space} from 1984 (J. Math. Phys.), Reinhard Werner developed a new phase space formalism which allowed for a joint harmonic analysis of functions and operators. Since his reasoning was mostly guided by motivations from the physical side of quantum mechanics, mathematicians ignored this highly interesting contribution for almost 35 years. Only in the last few years, interest in Werner's approach grew and actually yielded a number of interesting and relevant results in time-frequency analysis as well as in operator theory. The speaker, who has been working mostly on the operator theory side of quantum harmonic analysis (QHA), will try to describe the basic features of QHA and how they relate to problems in operator theory. After presenting some basics of the formalism of QHA, we will discuss one application of the audience's choice: Either a result in Fredholm theory, results in commutative operator algebras or a characterization problem of a certain important algebra appearing in QHA.
Posted December 2, 2024
Last modified February 7, 2025
Chian Yeong Chuah, Ohio State University
Marcinkiewicz Schur Multiplier Theory for Schatten-p class
The boundedness of Schur Multipliers plays an important role in the study of non-commutative harmonic analysis. In this talk, we provide a Marcinkiewicz type multiplier theory for the Schur multipliers on the Schatten p-classes. This generalizes a previous result of Bourgain for Toeplitz type Schur multipliers and complements a recent result by Conde-Alonso, Gonzalez-Perez, Parcet and Tablate. As a corollary, we obtain a new unconditional decomposition for the Schatten p-classes for p>1. Similar results can also be extended to the case of R^d and Z^d, where d>=2.
Posted January 20, 2025
Last modified February 10, 2025
Kabe Moen, University of Alabama
New perspectives on the subrepresentation formula
The classical subrepresentation formula establishes that a smooth function is bounded pointwise by the Riesz potential applied to its gradient. This fundamental inequality, combined with the mapping properties of the Riesz potential, leads to the celebrated Gagliardo-Nirenberg-Sobolev inequality, which plays a crucial role in analysis and partial differential equations. In this talk, we show a powerful extension of the subrepresentation formula to several prominent operators in harmonic analysis, including exotic cases such as rough singular integrals and spherical maximal functions. Additionally, we uncover some new structural properties of subrepresentation formulas, including an openness property and an equivalence with weighted Sobolev inequalities and isoperimetric inequalities.
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
3:30 pm – 4:30 pm TBA
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 January 16, 2025
Zi Li Lim, UCLA
TBA