Graduate Student Seminar - Ethan Acklesberg

February 13, 2020
Tuesday, February 18, 2020 - 5:15pm to 6:30pm
Cockins Hall 232
Graduate Student Seminar

Title: Structure and Randomness in Ergodic Ramsey Theory

Speaker: Ethan Acklesberg - The Ohio State University   

Abstract: Ramsey theory is based on the widespread phenomenon that “large” sets inevitably contain combinatorial structure on a smaller scale. A surprisingly useful method for proving results in Ramsey theory is via the study of recurrence in ergodic theory. We will give some indication of how combinatorial results can be interpreted as statements about dynamical systems (Furstenberg correspondence principle) and then outline the proofs of a few basic results in Ramsey theory. Namely, we will show that dense subsets of the integers (to be defined in the talk) contain (1) arithmetic progressions of length 3 (Roth’s theorem) and (2) patterns of the form {x, x+n^2} (Sárközy’s theorem). The key to both of these results—and many others in Ramsey theory—is a dichotomy between appropriately defined notions of “structure” and “randomness” governing the behavior of certain dynamical quantities.

Seminar Link

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