# 2023 Program for Women and Mathematics: Patterns in Integers: dynamical and number theoretic approaches

## Uhlenbeck Lecture Course: Ergodic Ramsey Theory

Abstract: A famous theorem of Szemeredi from 1975 states that any subset of positive density in the integers contains arbitrarily long arithmetic progressions. In 1977 Furstenberg gave an ergodic theoretic proof of Szemeredi’s theorem. Furstenberg observed that combinatorial statements about patterns in the integers correspond to multiple recurrence questions in ergodic theory. This gave rise to the field of Ergodic Ramsey theory, which centers around proving Ramsey type results using ergodic theoretic techniques (some such results have not alternative proof to this day!). The course will introduce the participant to some ideas in ergodic Ramsey theory and also to connections with other approaches to Ramsey type problems including the circle method, which will be introduced in the second course.