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 the precise mapping we call these functions Hecke vector-forms. In a preprint, the mapping of my dissertation becomes the foundation of an analytic bijection between quasiautomorphic forms and Hecke vector-forms. Thus, all quasiautomorphic forms are, fundamentally, just automorphic forms when observed in the correct light.
My research concerns the development of a fully explicit theory of these vector-forms, not just over the Hecke triangle groups, but also the Schwarz triangle groups, generally (compact triangle groups, most importantly). This analytic formulation will generalize much of the standard classical tools such as the Hecke operator and the Petersson inner product. Soon I will post a preprint where the story is laid out in full for Hecke groups.
To clarify the aims of the program, the Schwarz triangle groups have been largely ignored by practitioners of automorphic forms except for the well-known exceptions; however, as a class they are essential to a complete understanding of algebraic curves — we now know this because Belyi’s theorem has a triangle group formulation. For more information on the connections between triangle group geometry and algebraic curves, see the fascinating book.
Michael Andrew Henry 