Abstract: Sums of Kloosterman sums against other functions often arise in analytic number theory problems, but it is difficult to prove bounds when the length of the sum is below the square root of the modulus. In joint work with Emmanuel Kowalski and Philippe Michel, we proved a bound on bilinear forms in Kloosterman sums where the length is just below this range. This has applications to moments of L-values of modular forms and to the level of distribution of Eisenstein series. I will explain this result and discuss possible generalizations.
Abstract: The Riemann hypothesis over function fields is a theorem of Weil, with a more general form due to Deligne. This raises the hope of answering more precise questions on the zeroes of L- functions, such as the distribution of the zeroes in a suitable family of L-functions. Powerful results on this have been proved, starting with the book of Katz and Sarnak, but the theory is not yet complete, and it is an area of active research. I will survey the history, explain recent developments, and discuss what is in reach for the future.
Abstract: The breakthrough work of Marcus, Spielman, and Srivastava showed that every bipartite Ramanujan graph has a bipartite Ramanujan double cover. Chris Hall, Doron Puder, and I generalized this to covers of arbitrary degree. I will explain the proof, with emphasis on how group theory and representation theory are useful for this problem.
Abstract: I will explain the analogy between trace functions over finite fields defined by exponential sums and certain classical functions on the complex numbers defined by integrals of exponentials. There are close analogies, largely due to Katz, that sometimes allow one to guess results in one domain from results in the other. For instance, many important properties of Kloosterman sums are related to facts about Bessel functions. I will explain some of these correspondences, and how to use them to understand exponential sums.
Abstract: In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.