UBC Math Department Colloquium: Didier Lesesvre

In 1671, Bachet conjectured that every natural number can be written as sum of four squares. A century of efforts and ideas was needed for Lagrange to reach a proof, and almost another century for Jacobi to quantify precisely this result. Today, variations of this elementary questions remain far beyond reach.

This adventure will be my excuse to introduce modular forms, which are one of the numerous avatars of automorphic representations. These objects are very rich but remain mysterious, and there is little hope to understand them precisely with current methods. Remains the option of studying them on average, embedding them in families of automorphic forms, and using tools from harmonic analysis on groups to quantify the behavior of these automorphic representations: this is the entrance to the realm of arithmetic statistics.

I will take the time to introduce modular forms, automorphic representations, motivations and statistical challenges. The central tool in this approach is the trace formula, which builds a bridge between spectrum of operators, geometry of a varieties and arithmetic of groups. This formula, fundamental and fruitful in many contexts, will be smoothly introduced, commented and proven in simple cases, and will allow to answer some of our original questions.