Harmonic analysis in Australia

Workshop: Harmonic Analytic Connections

Date: May 26-31, 2024

Organizers: Ji Li, Pierre Portal, Alexandria Rose and Po-Lam Yung

The workshop is a part of the Special Year in Harmonic Analysis at the Mathematical Sciences Institute, Australian National University. It is held and partially supported by MATRIX at Creswick, Australia. The main webpage for the workshop can be found here.

Title and Abstracts:

Hong Wang (NYU)
Incidence estimate and oscillatory integrals

Abstract: We will discuss some recent multi-scale method on incidence estimate for tubes and its application to oscillatory integrals. The multi-scale method was developed by Keleti-Shmerkin, Orponen, Shmerkin, etc.

Camil Muscalu (Cornell)
A new approach to the Fourier Extension Problem for the paraboloid [Slides]

Abstract: The plan of the talk is to describe a new approach to the so-called Restriction Conjectures, that Itamar Oliveira and I have developed recently. Without entering into details, this new point of view allows one to prove that (essentially) all the relevant conjectures (linear or multi-linear) are true, provided that one of the functions involved has a tensor structure.

Alex Iosevich (Rochester)
Analytic, combinatorial and number theoretic manifestations of the Fourier Uncertainty Principle [Slides]

Abstract: The Fourier Uncertainty Principle says, roughly, that both the function and its Fourier transform cannot be simultaneously localized. This idea is ubiquitous in many areas of analysis and related areas, from restriction/decoupling inequalities in harmonic analysis to signal recovery in electrical engineering. In this talk, we are going to expose certain aspects of the Fourier Uncertainty Principle, with emphasis on the diversity of ideas and applications.

Carlos Perez (University of the Basque Country and BCAM)
Connections between Harmonic Analysis and the self-improving property of the oscillation of a function [Slides]

Abstract: In this lecture, we will show connections between Harmonic Analysis and certain fundamental estimates, such as the Poincar\'e-Sobolev, Trudinger or John-Nirenberg inequalities. The basic theme is the self-improving property of generalized Poncar\'e type inequalities. Through this exploration, we will demonstrate how these connections allow us to avoid the dependence on Potential Operators. This approach is more flexible but also produces more precise estimates, particularly when dealing with singular measures. We will outline some old results which can be improved in modern ways which include fractional type results improving some celebrated results by Bourgain-Brezis-Mironescu.

Zane Li (North Carolina State University)
Comparing and contrasting decoupling and efficient congruencing [Slides]

Abstract: Vinogradov's Mean Value Theorem (VMVT) was proven by Bourgain-Demeter-Guth in 2015 as a corollary of decoupling for the moment curve. Roughly parallel in time, Wooley was developing the theory of efficient congruencing which culminated in the nested efficient congruencing proof of VMVT in 2017. We compare and contrast both decoupling and efficient congruencing methods and justify how in some sense efficient congruencing is some sort of "p-adic decoupling".

Jongchon Kim (City University of Hong Kong)
Weighted decoupling inequalities and the convergence of Bochner-Riesz means

Abstract: In this expository talk, I will discuss some weighted l^p-decoupling inequalities recently discovered by Gan and Wu and its application to the problem on the almost everywhere convergence of the Bochner-Riesz means for L^p functions for p<2.

Ruixiang Zhang (UC Berkeley)
How to produce curved Kakeya sets that are small [Slides]

Abstract: This talk concerns curved Kakeya sets coming from the study of H\"{o}rmander type operators. When the "position function" is good enough, there is a conceptual way to approach the problem of whether curved Kakeya sets can be small. I will talk about this viewpoint and present some results and questions, explained by examples.

Participants: