Time:  4:305:50 (NONSTANDARD TIME) 
Room: 
Doherty Hall A310 (NONSTANDARD PLACE)

Speaker: 
Bryna Kra Department of Mathematics Northwestern University 
Title: 
Connections between ergodic theory and additive combinatorics

Abstract:  Much recent work in ergodic theory has been motivated by interactions with combinatorics and with number theory. A striking example is Szemerédi's Theorem, which states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon after Szemerédi's proof, Furstenberg gave a new proof using ergodic theory. This opened new questions in ergodic theory, and developments in ergodic theory, in turn, have lead to breakthroughs in additive combinatorics. It turns out that algebraic constraints (nilsystems) play a key role in understanding these phenomena, both in additive combinatorics and in ergodic theory. I will give an overview of the role of nilsystems in the recent developments, explaining the beginnings of a theory of higher order Fourier analysis that is a tool for addressing open problems in the area. 