Computational Aspects of Algebraic Geometry, Automorphic Forms, and Number Theory

Some of the most exciting and challenging mathematics is happening at the interface of three areas: Algebraic Geometry, Automorphic Forms and Number Theory. The hub of this is the Langlands Program. This is a vast complex of conjectures that will keep mathematicians busy for decades. Theoretical progress has been made recently, for instance the solution to Serre’s conjecture, the proof of the Fundamental Lemma, and the proof of the Sato-Tate conjecture. These conjectures predict often very concrete information about, for example, solutions to polynomial equations with integer coefficients. In turn, these conjectures suggest interesting avenues of research within related domains, and new territory to explore. In mathematics, often this exploration takes the form of experimentation with examples. In fact the examples are often interesting in their own right. A good instance of this is the theory of Calabi-Yau varieties. These varieties have played an important role in physics, but also the study of their arithmetic has been also fruitful. The zeta-functions of these varieties have even been studied by physicists (see, e.g., [1]) We propose to have a workshop devoted to topics at the interface of these three areas, with an emphasis on computation.