RG Analysis and Partial Differential Equations

Summer term 2024

S4B1 - Graduate Seminar in Analysis

Polynomial Methods in Incidence Geometry, Harmonic Analysis and Number Theory


  • Dr. Rajula Srivastava
  • email: rajulas (at) math (dot) uni-bonn (dot) de.
  • Dates


    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: 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).


  • 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)


    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.


    Talk preparation