Additive Combinatorics is a modern mathematical field between Number Theory and Combinatorics which uses intesevely methods from Harmonic Analysis, Graph Theory, Ergodic Theory, Probability Theory, Algebraic Geometry, Topology and Geometry of Numbers.

In our course we discuss modern and classical results in the area as well as with the connection of Additive Combinatorics with other mathematical fields.

The central theme of the first semester is the affine sieve of Bourgain–Gamburd–Sarnak. Also, if time permits, we will discuss an arithmetic version of Erdos–Hajnal conjecture.

In the second semester we will continue studying of applications of non-commutative methods to some problems of Additive Combinatorics. Also, we will discuss the connection between GCD-sums and the uniform distribution of sequences as well as with the additive energies.

Sujets de travaille:

1. GCD-sums and its connection with uniform distribution of sequences (poissonian, disstribution of the minimal distance and so on) and additive energies. The connection with Dirichlet polynomials and zeta-functions.
2. Additive and multiplicative Markov chains over F_p. Chung-Diaconis-Graham process. The connection with analytical methods, sum-products and Incidence Geometry.
3. Erdos-Hajnal conjecture.
4. Introduction to the determinant method.

Financial support. The seminar is supported by the Simons Foundation and the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2019-1614).

Lectures (.zip)

Seminar organizer
Shkredov Ilya Dmitrievich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center