Seminar

In this talk, which I hope will be of interest to participants from both programmes, I will discuss some joint work with J. Wright, formulating certain harmonic analysis problems over Z/NZ. I will also briefly survey how various related classical number theoretic results (for instance, Hua's exponential sum estimates) can be naturally approached from a harmonic-analytic perspective. 



Recently, substantial progress has been made on several major open problems in Euclidean harmonic analysis inspired by first considering discrete models formulated over finite fields. Finite fields do not encapsulate all aspects of analysis over R, however, and the lack of any non-trivial notion of scale in the former leads to divergence between the two theories.



One approach to modeling multiple scales in a discrete setting is to work over rings of integers Z/p^kZ (or, more generally, Z/NZ for composite N). In this case, each `scale' corresponds to a possible number of divisors and the resulting theory closely parallels that of the Euclidean case.