# Special Year on Dynamics, Additive Number Theory and Algebraic Geometry

During the 2022-2023 academic year, the School will have a special program on

Applications of Dynamics in Number Theory and Algebraic Geometry. Tamar Ziegler of the Hebrew University in Jerusalem will be the Distinguished Visiting Professor.

Confirmed participants include V.Bergelson, J.Draisma, J.Maynard, S.Peluse, T.Tao.

The special year will focus on (still-evolving) interfaces between ergodic theory, additive combinatorics, multiplicative number theory, and algebraic geometry.

In the past 20 years there have been spectacular interlacing advancements in ergodic theory, analytic number theory and additive combinatorics, leading to the resolution of long standing conjectures. A well known example is the Green-Tao theorem on arithmetic progressions in primes. A more recent example is the proof of the logarithmic Sarnak conjecture for uniquely ergodic systems; this proof combines structure theorems from dynamics and additive combinatorics with the recent understanding of the behavior of multiplicative functions in short intervals. One focus of the special year will be to further advance the interaction between these fields.

Another goal of the program will be to advance the interplay between additive combinatorics, theoretical computer science and algebraic geometry over rings with many variables. In recent years there is growing interest in stability phenomena in algebraic geometry, specifically in properties of polynomial rings that are stable in the number of variables (e.g. Stillman conjecture). Problems of a similar nature were independently studied by researchers in additive combinatorics in relation to Ramsey questions in finite field geometry. The two contexts are linked via model theory. The program will bring together researchers on both sides to explore this connection further.