GROUPS, ARITHMETIC AND ALGEBRAIC GEOMETRY SEMINAR - Sidon sets, Larsen's alternative, exponential sums, and random graphs

A Sidon set is a subset S of an abelian group A with the property that elements of A cannot be sums of two elements of S in more than one way. Larsen's alternative is a criterion to decide whether a compact subgroup K of the group of unitary matrices is as large as possible in terms of the fourth moment of K. I will explain how to construct Sidon sets coming from algebraic geometry, and how Larsen's alternative allows us to prove explicit quidistribution statements for some exponential sums over finite fields supported at those Sidon sets. Building on this, we can construct families of graphs that, on the one hand, have eigenvalues that are asymptotically distributed according to the Sato-Tate measure, but on the other hand are very different from random graphs, for example in that they do not contain K_{2, 3}as a subgraph. This is a joint work with Arthur Forey, Emmanuel Kowalski, andYuval Wigderson. 

(30 minutes general talk, followed by 45minutes of more specialized talk)