SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Brian Cook
Abstract. For a fixed integer $d>1$, Waring's problem asks whether or not there exists an integer $k$ such that every positive integer can be written as a sum of $k$ $d$th powers of integers. For example, a result of Lagrange states that every positive integer is a sum of four squares. The problem, as stated, was solved by Hilbert around the turn of the 20th century.In the 1930's, L.K. Hua extended the work of Vinogradov on the ternary Goldbach conjecture, now known as Vinogradov's Theorem, to give an analogue of the result of Hilbert which is set in the primes. For example, Hua has shown that every sufficiently large integer is the sum of at most nine squares of primes. The main purpose of this talk is to discuss some recent work, joint with T. Anderson, K. Hughes, and A. Kumchev, on an ergodic analogue of the work of Hua. The focus shall be mostly on the case of squares, and here the main ingredient is an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger.