Patterns in Integers: dynamical and number theoretic approaches
Uhlenbeck Lecture Course: Ergodic Ramsey Theory
Lecturer: Tamar Ziegler, IAS/Einstein Institute of Mathematics, Hebrew University
Teaching Assistant: Adi Gluksam, Northwestern University
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.
Terng Lecture Course: The Circle Method
Lecturer: Lillian Pierce, Duke University
Teaching Assistant: Rena Chu, Duke University
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.