Peter Sarnak, Professor, School of Mathematics. Through the works of Fermat, Gauss, and Lagrange, we understand which positive integers can be represented as sums of two, three, or four squares. Hilbert's 11th problem, from 1900, extends this question to more general quadratic equations. Its complete solution relies on recent advances in related fields as well as many developments over the years. Professor Sarnak explains some of these as well as select far-reaching conjectures that the problem has inspired.