Trevor Wooley, University of Bristol
“A translation-invariant perspective on arithmetic (and) harmonic analysis”
We learn about translation-invariance at mathematical birth. Thus, given a three-term arithmetic progression, such as 10, 20, 30, we learn that one can obtain another such progression by shifting every entry by the same integer, as in the triple 17, 27, 37. Very recently, a method has emerged that ap- plies non-linear translation-invariant equations to analyse Fourier series having polynomial arguments (such as Weyl sums) – the p-adic or congruence based version of this method is known as efficient congruencing, while the real interval based version is the decoupling method of Bourgain, Demeter and Guth. There are many circumstances in which this method achieves the best possible estimates, with consequent applications of Diophantine type such as Waring’s problem, equidistribution of poly- nomial sequences modulo one, and discrete Fourier restriction problems. In this talk, while we outline the underlying ideas, we emphasise applications. We will discuss and speculate concerning as many different problems as time permits.
MC 5501
Refreshments will be served in MC 5403 at 3:30 p.m. Everyone is welcome to attend.