Representations and unrepresentations by sums of powers

We report on recent and on-going work joint with Joerg Bruedern concerning problems involving the representation of integer sequences by sums of powers. Our new tool is an upper bound for moments of smooth Weyl sums restricted to wide major arcs. This permits progress to be made on Waring's problem, explicit versions of Freiman's theorem, and other problems involving mixed sums of powers and primes. Recent progress addresses unrepresentation theory and problems involving r-free numbers.