Beyond Fourier Analysis
Monday January 28, 2008
09:30AM - 10:30AM
Speakers:
Bryna Kra
Abstract:
Much recent work in ergodic theory has been motivated by interactions with combinatorics and with number theory. A striking example is Szemeredi's Theorem, which states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon after Szemeredi'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. While Fourier analysis is useful for understanding some patterns (such as arithmetic progressions of length 3), it does not suffice for understanding general patterns. It turns out that algebraic constraints (nilsystems) play a key role in understanding the more complicated 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 the main tool for addressing open problems in the area.
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