Many Diophantine problems involve arithmetically defined points on a continuous manifold, These can often be described by number-theoretically determined discrete symmetries of a homogeneous geometric space. An important example arises in the study of positive definite integral quadratic forms. This is a classical area of research going back to Euler, Gauss, Voronoi etc., but one that is still very relevant (e.g. the LLL algorithm) and remains an active area of research.

Using automorphic forms - functions invariant under these discrete symmetries - such problems can be connected to geometry and representation theory, offering new perspectives. This approach has led to significant results and inspired numerous breakthroughs recognized with Abel Prizes and Fields Medals. Notable examples include Wiles’ proof of Fermat’s Last Theorem, Margulis’ construction of expander graphs, Viazovska’s work on optimal lattices, and advancements in the Langlands program, though the list is extensive.

My research focuses on the role of automorphic forms in solving various problems in arithmetic and geometry.