Hello and welcome to my professional site!
I recieved my PhD from TU Graz in Austria. In my dissertation project (advised by Peter J. Grabner), I define an analytic mapping from an arbitrary quasiautomorphic form (modulo any Hecke triangle group) to a vector-valued automorphic form: under this precise mapping we call these functions Hecke vector-forms. In a recent preprint, I extended this mapping to a bijection between quasiautomorphic forms and Hecke vector-forms. Thus, all quasiautomorphic forms are fundamentally just automorphic forms under the correct light. This has many deep implications, but my particular interests are in number theory, enumeration, and mathematical physics.
In general, my research concerns developing a fully explicit theory of these vector-forms — but not just over the Hecke triangle groups. The goal is a functional theory over all Schwarz triangle groups. Indeed, for the analytic formulation I have in mind (extending from Riemann’s mapping theorem), this would generalize all the standard classical tools such as the Hecke operator and the Petersson inner product to significantly more settings than currently allowed.
This is not generalization for its own sake, but because the Schwarz triangle groups (and their subgroups) are essential to a complete understanding of algebraic curves; this we know because Belyi’s theorem has a triangle group formulation. This means extending the automorphic forms framework beyond just arithmetic triangle groups may prove vital in a future studies of general algebraic curves. For more information on the connections between triangle group geometry and algebraic curves, see the fascinating book here.
Michael Andrew Henry 