Please note special day (Friday).  We will start by motivating some problems in discrete harmonic analysis by discussing their Euclidean counterparts. Then I will discuss Stein's spherical maximal function, Magyar's discrete version and the ideas behind Magyar-Stein-Wainger's theorem (proving L^p boundedness). We will pay particular attention to the synthesis of ideas from harmonic analysis and analytic number theory. I will then discuss higher degree versions and lacunary versions. I will conclude with applications to ergodic theory and combinatorics. If time permits, we will discuss the connection to restriction theorems.