Posts

2023-2024

The regular time and place for the seminar is Tuesday at 3:30 – 4:30 pm in Math Tower 154. If you are interested, please contact Liding Yao. If you are interested to topics of mathematical analysis related to operator theory, take a look to the Analysis and Operator Theory Seminar.

date speaker institution title
Sep 5 Alex McDonald OSU The Newhouse gap lemma and patterns in products of Cantor sets (part I)
Sep 12 Alex McDonald OSU The Newhouse gap lemma and patterns in products of Cantor sets (part II)
Sep 19 Adam Christopherson OSU Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3
Oct 20 Ben Bruce (on Zoom) UBC Hausdorff dimension and patterns determined by curves
Oct 31 Gennady Uraltsev University of Arkansas Multilinear and uniform bounds in harmonic analysis
Nov 7 Terence Harris UW-Madison Horizontal Besicovitch sets of measure zero and some related problems
Nov 14 Eyvindur Ari Palsson Virginia Tech A restricted Falconer distance problem
Nov 28 Robert Fraser Wichita State Oscillatory integrals arising from algebraic number fields

Alex McDonald

Title: The Newhouse gap lemma and patterns in products of Cantor sets

Abstract: Many important problems throughout mathematics can be summarized as follows: how large must a set be in order to ensure that it exhibits some type of structure?  A classic example of importance in geometric measure theory and harmonic analysis is the Falconer distance problem, which asks how large the Hausdorff dimension of a compact set must be to ensure it determines a positive measure worth of distances.  More generally, one may ask for an abundance of more complicated point configurations.  There has been considerable recent progress on variants of these problems where Hausdorff dimension is replaced by a quantity called thickness, which provides an alternative way to quantify the “size” of a compact set.  In these talks, I will give an overview of thickness from the ground up, and discuss some applications to finding point configurations in Cantor sets.

Adam Christopherson

Title: Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3

Abstract: In this talk, we give a complete characterization of the weak-type regularity of the Bergman projection on rational  power-generalized Hartogs triangles in C^3. In particular, we show that the Bergman projection satisfies a weak-type estimate only at the upper endpoint of L^p boundedness on both of these domains. A similar result has been observed by Huo-Wick and Koenig-Wang for the classical Hartogs triangle and punctured unit ball, respectively. This work is joint with K.D. Koenig.

Ben Bruce

Note: Ben will give a Zoom talk on Zoom on Oct 20 11:30- 12:25, during the class Math 8210 at Scott Lab E103. Zoom link is here.

Title: Hausdorff dimension and patterns determined by curves

Abstract: In this talk, I will discuss joint work with Malabika Pramanik on the problem of locating patterns in sets of high Hausdorff dimension. More specifically, suppose Γ is a smooth curve in Euclidean space that passes through the origin. Is it true that every set with sufficiently high Hausdorff dimension must contain two distinct points x,y such that x-y \in Γ? We showed that if Γ is suitably curved then the answer is yes, while for certain flat curves the answer is no. This generalizes work of Kuca, Orponen, and Sahlsten, who answered this question affirmatively when Γ is the standard parabola in R^2.

Gennady Uraltsev

Title: Multilinear and uniform bounds in harmonic analysis

Abstract: See the pdf here.

Terence Harris

Title: Horizontal Besicovitch sets of measure zero and some related problems

Abstract: We show that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are discussed for the SL_2 Kakeya maximal function.

Eyvindur Ari Palsson

Note: Eyvi’s talk will be at Baker Systems Engineering, BE394. Also he will give a pre-talk on Nov 13 11:30-12:25 at E103, during the class Math 8210.

Title: A restricted Falconer distance problem

Abstract: The Falconer distance problem, a continuous analogue of the celebrated Erdos distance problem asks: How large does the Hausdorff dimension of a Borel set, in the plane or higher dimensions, need to be to ensure that the Lebesgue measure of its distance set is positive? This question has seen much progress in recent years, yet the conjectured threshold remains open. After a quick introduction of this problem I will give an overview of a number of variants leading to a formulation of a restricted Falconer distance problem with a particular example being a diagonally restricted distance set.

Robert Fraser

Title: Oscillatory integrals arising from algebraic number fields

Abstract: We introduce a new class of real-variable oscillatory integrals arising as the trace of a polynomial in the algebra (R ⊗_Q K) where K is an algebraic number field. We prove bounds for such oscillatory integrals that generalize a classical result of Arhipov, Cubarikov, and Karacuba for real polynomials and a recent result of Wright for complex polynomials.