Title: Uniform spectral gap in number theory and beyond
I’ll begin by discussing Selberg’s eigenvalue conjecture. This is an analog of the Riemann hypothesis for a special family of Riemann surfaces that feature heavily in number theory, for example in Wiles’ proof of the Taniyama-Shimura conjecture. I’ll explain how in the last 10 years, number theorists have had to turn to Anosov dynamics to obtain the approximations to Selberg’s conjecture that became relevant to emerging ‘thin groups’ questions about Apollonian circle packings and continued fractions. Then if I have time, I’ll explain how I have proved an extension of Selberg’s 3/16 theorem to higher genus moduli spaces, and point out some interesting ingredients of the proof.
Part of the talk is based on joint work with Oh and Winter and with Bourgain and Kontorovich.