Mathematical Institute of the University of Bonn

RG Analysis and Partial Differential Equations

Summer term 2024

S4B1 - Graduate Seminar in Analysis

Polynomial Methods in Incidence Geometry, Harmonic Analysis and Number Theory

Instructor

Dr. Rajula Srivastava

email: rajulas (at) math (dot) uni-bonn (dot) de.

Dates

• Preliminary Meeting Time: 07.02.2024, 14.15-15.15. Room: SR 0.008.
• If you are interested in the seminar but can't attend the preliminary meeting, please send me an email.
• Usual Meeting Time and Place: Thursdays, 14-16 (c.t.), Zeichensaal (Wegelerstr. 10).

Content

In the last two decades, a number of long standing open problems in incidence geometry, harmonic analysis and number theory have been solved using polynomials. The proofs, a number of which are extremely short and elegant, often took the mathematical communities by surprise, since the questions they answered had a priori got nothing to do with polynomials.

In this course, we will look at some of these problems and their solutions using Polynomial Methods. These include:

• The finite field Kakeya problem, inspired by the (still open) Kakeya conjecture from Euclidean harmonic analysis.
• The Joints problem and the Szemerédi–Trotter theorems from Incidence Geometry, dealing with the number of incidences of lines in space or the plane.

To fully appreciate the difficulty of these problems, we will compare their current proofs (employing the polynomial method) with the previous attempts without polynomials. This will be the content of Topics 1-6.

Next, through Topics 7-10, we will focus on how incidence geometry and combinatorics arise naturally in the context of several problems from Euclidean Harmonic Analysis, especially the aforementioned Kakeya conjecture. Another aim will be to understand the scope and limitations of the polynomial method when applied to these problems.

Topics 11-12 deal with the application of the Polynomial Method to Diophantine Approximation (Thue’s Theorem).

Reference

We will mostly follow the book Polynomial Methods in Combinatorics by Larry Guth.

Please do not buy the book unless you absolutely want to. There is a reference copy in the Mathematics library. If you cannnot access it for your talk, please email me.

Topic List (with Presenters)

Prerequisites

There are no rigid prerequisites, but an interest in at least one of the areas among Harmonic Analysis, Incidence Geometry and Number Theory is desirable. We will bypass/black box the few results required from Algebraic Geometry.

Timetable

• April 11: Topic 1 (Handout by Piro Manco)
• April 18: Topic 2 (Handout by Tejas Ramesh)
• April 25: Topic 6 (Handout by Daniel Sirola)
• May 2: Topic 7 (Handout by Izabela Mandla)
• May 9: No seminar (holiday)
• May 16: Topic 8 (Handout by Shao Liu)
• May 23: No seminar (Pentecost Break)
• May 30: No seminar (holiday)
• June 6: Topic 9 (Handout by Dennis Wollgast)
• June 13: No seminar
• June 20: Topic 10 (Handout by Jianghao Zhang)
• June 27: Topic 11 (Handout by Edward Young)
• July 4: Topic 12 (Handout by Lin Feng)
• July 11: No seminar
• July 18: Topic 4 (Handout by Johannes van Rensen)

Talk preparation

• Please set up a meeting with me least two weeks before your presentation date to discuss your talk.
• The talk must be supplemented by a handout summarizing the topic assigned. Please being the handout with you to the meeting mentioned above.