Joint PM/CO Grad Colloquium

Mani Thamizhazhagan, Department of Pure Mathematics, University of Waterloo

"On the interplay of harmonic analysis, combinatorics, additive number theory and ergodic theory"

Arithmetic combinatorics, or additive combinatorics, is a fast-developing area of research combining elements of the fields in the above title. Its conceivably best-known result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the long-standing conjecture that primes contain arbitrarily long arithmetic progressions. The purpose of this colloquium talk is to sketch a line of research on a wider, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the Green-Tao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. Some combinatorial theorems on intersecting lines and circles give life to a major result in partial differential equations. An unforeseen argument directs harmonic analysis towards additive number theory, with consequences that could have hardly been anticipated.