Title:     Recent Results in Discrete Harmonic Analysis

Speaker:     Dr. Angel Kumchev, Professor. Department of Mathematics

Abstract:     In this lecture, I will give a brief overview of some recent developments in discrete harmonic analysis. We will be concerned with so-called "discrete maximal functions.''  These are discrete analogues of the classical spherical maximal operator.  A classical result of E. Stein establishes that this operator is bounded on L^p(R^d) for p > d/(d-1).  Stein's theorem has been generalized to averages over dilates of other surfaces and has inspired a great deal of work in harmonic analysis. In recent years, driven by applications to ergodic theory, research on such topics has expanded to maximal operators for averages over discrete sets of arithmetic interest, such as averages over the integer points on spheres of integer radii in R^d.  In this talk, I will present a brief history of the subject and highlight some work that I have done jointly with T. Anderson, B. Cook, K. Hughes, and E. Palsson on such maximal functions, including during my recent sabbatical.  As this talk is intended for a general audience, with some bend towards number theory, I will focus on explaining the harmonic analysis roots of the problems, the connections to number theory, and the role of number-theoretic methods.