# Harmonic Analysis and Analytic Number Theory

Dual Trimester Program

May 3 - August 20, 2021

Organizers: Valentin Blomer, Farrell Brumley, Philip Gressman, Marina Iliopoulou, Lillian B. Pierce

**Online participation in the events of the trimester is available worldwide, including online participation in the two summer schools and the two seminar series. Please register to attend the events online by following the appropriate link on the pages of these activities.**

Analytic Number Theory is a rich and highly active field, with core areas such as the study of the distribution of primes, Diophantine equations, L-functions and automorphic forms, and also connections to algebraic geometry, the Langlands program, arithmetic statistics, arithmetic geometry, and dynamics. Similarly, harmonic analysis is a rich and highly active field, with core areas such as singular integral operators, oscillatory integrals, restriction and Kakeya problems, time-frequency analysis, and also connections to PDEs, geometric measure theory, incidence geometry, and arithmetic combinatorics.

While connections between Analytic Number Theory and Harmonic Analysis have been visible for many years, very recent powerful observations are opening up striking new approaches and new open questions, which this joint trimester aims to develop. Topics that will be explored at this exciting interface include: the methods of decoupling and efficient congruencing, which have resulted in tour-de-force proofs of the Vinogradov Mean Value Theorem; polynomial methods, which exhibit stunning versatility in addressing problems ranging from exponential sum bounds and counting points on varieties to incidence geometry, the Kakeya problem, and restriction estimates; applications of the circle method to geometric settings, harmonic analysis and ergodic theory; oscillatory integrals both from a geometric perspective in harmonic analysis, and also from the perspective of microlocal analysis and applications to trace formulae and automorphic periods and L-functions; connections between modular forms and discrete geometry, such as the breakthrough resolution of sphere-packing problems in dimensions 8 and 24.

The HIM trimester in Harmonic Analysis and Analytic Number Theory will provide a focused research environment for mathematicians spanning both fields, and will feature seminar series, open problem days, and a summer school, which will stimulate cross-pollination between the fields. We welcome applications from diverse mathematicians, of all career stages, in these areas.

**Those planning to participate include:** Jörg Brüdern, Tony Carbery, Mike Christ, Ciprian Demeter, Shaoming Guo, Roger Heath-Brown, Emmanuel Kowalski, Akos Magyar, Simon Marshall, Lilian Matthiesen, Jasmin Matz, Philippe Michel, Ritabrata Munshi, Malabika Pramanik, Maksym Radziwill, Andreas Seeger, Kannan Soundararajan, Betsy Stovall, Nicolas Templier,

Terence Tao, Maryna Viazovska, Hong Wang, Trevor Wooley, Jim Wright, Matt Young

Given the current situation with COVID-19, the ongoing hygienic restrictions, difficulties in obtaining visas and further obstacles for travelers, we will see to it that parts of the program may also be attended virtually (non-funded). Please note that the program itself is an on-site event and physical presence is vital.

**Activities during the trimester program:**

The program will include 2 Schools addressed to PhD students and postdocs and 2 major lecture series:

- Hausdorff School: The Circle Method (May 10-14, 17-21, 2021)
- Summer School: Polynomial Methods (June 7-17, 2021)
- Seminar Series: Arithmetic Applications of Fourier Analysis
- Seminar Series: Harmonic Analysis from the Edge