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

Abstract: This series will invite participants into the beautiful world of the circle method. This method, which combines both arithmetic and analytic insights, originated 100 years ago in work of Hardy and Ramanujan, in their study of the partition function. It was then more fully developed by Hardy and Littlewood in the study of Waring’s problem, which asks how many ways a given integer may be expressed as a sum of s perfect k-th powers. We will introduce the mechanics of the circle method in the setting of Waring’s problem. Then we will explore the relationship to Ergodic Ramsey Theory by applying the circle method to address questions like the following: must every set of integers with positive density contain a 3-term arithmetic progression? The series will conclude by giving a “world tour" of mathematicians currently investigating problems via the circle method.