The main thrust of the research project is the continuation of the joint work of the PI with T. Hausel and E. Letellier on the geometry of certain character varieties, the moduli space of representations of the fundamental group of a Riemann surface to a reductive algebraic group. These spaces have a rich geometry. In other incarnations they parametrize certain Higgs bundles on the surface, a space which arose in connection to theories in Physics. The main conjecture of the PI and his collaborators describes the mixed Hodge structure of the character varieties and involves the Macdonald polynomials of combinatorics. In a separate project the PI will continue to investigate the theoretical and computational aspects of the L-functions attached to hypergeometric motives.
It is hard to predict how current pure Mathematics will impact life outside its discipline. But overall, it largely does sooner or later. Arithmetic geometry in particular, has become crucial in a world of ever increasing digital communication and data manipulation. Geometry over finite fields, for example, once a subject of specialists, is now needed for many widely used algorithms for encryption and electronic transmission of information. In this project the PI uses finite fields in the other direction: as a probe to study the geometry of some spaces of relevance in Mathematics and Physics.