Advanced Course on Modern aspects of Fourier analysis

Measures with pure point support and spectrum

Felipe Gonçalves

IMPA

In this series of three lectures we will give an overview of an unique and intriguing phenomenon that permeates different branches of mathematics and often serves as a key analytical entry point to many problem-solving strategies: Summation formulas for the Fourier transform. In modern lingo: When can a measure have pure point support and pure point spectrum? We will present many examples of such measures (and formulae), tracing their evolutionary lineage from ancient to modern. These creatures thrive in mathematical habitats such as: number theory (prime summation and trace formulas), discrete geometry (lattices and model sets), harmonic analysis (Fourier reconstruction formulae and modular forms). We will then discuss some recent lineage classification results, explore the role of almost periodicity and present some “Darwin hawkmoth” conjectures.

Counterexamples to pointwise convergence questions in dispersive PDE’s

Lillian Pierce

Duke University

In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation in n-dimensional Euclidean space (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated question was recently resolved by a combination of two results: one by Bourgain, whose counterexample construction proved a necessary condition, and later a complementary result of Du and Zhang, who proved a sufficient condition. 

Bourgain’s counterexample was particularly interesting for two reasons: first, it generated a necessary condition that contradicted what people had expected, and second, it seemed to gain its power from simple properties of Gauss sums. In this lecture series we will explain a way to derive Bourgain’s counterexample from first principles, then we will describe a more flexible way to think about such constructions, which leads to many more classes of counterexamples.

Summation formulas, modular forms, and applications

Danylo Radchenko

University of Lille

The Poisson summation formula is one of the oldest and most useful tools in Fourier analysis, having widespread and often quite profound applications in many areas of mathematics, from time-frequency analysis and crystallography to coding theory and number theory. In this course I will talk about this and other summation formulas, both classical and modern, for the Fourier and other integral transforms, with a focus on the interplay between summation formulas, modular forms, and Dirichlet series. I will also discuss some of the applications of these identities in geometry and number theory.

Some tools in Fourier Restriction, Kakeya and Weighted Restriction

Ruixiang Zhang

UC Berkeley

We will introduce three interconnected topics, namely Fourier Restriction, Kakeya and Weighted (Fourier) Restriction, and briefly talk about some connections to problems in a few related math fields. After that, we will cover some tools that recently have been useful for making progress on the key problems.