He works in algebraic and arithmetic geometry, automorphic forms, and the relation between number theory and other fields, such as logic, noncommutative geometry and mathematical physics.

In his 1997 PhD, he studied the distribution of zeros of Eisenstein series for function fields, with applications to supersingularity of Drinfeld modules. In connection with this gave criteria for the existence of rational 2-power torsion points on Jacobians of hyperelliptic curves over finite fields. He also applied the theory of Castelnuovo-Mumford regularity to rings of Drinfeld modular forms.
With Fumiharu Kato and Aristides Kontogeorgis, he has worked on orbifold curve uniformization over fields of positive characteristic. This includes a sharp upper bound on the number of automorphisms of a Mumford curve in any characteristic, the solution of the "Hurwitz Group" problem in this situation, a study of the analytic equivariant deformation theory for Mumford curves, and a comparison of this to the algebraic theory.
With Fumiharu Kato, Ariane Mezard and Jakub Byszewski, he has been working on a long term project to completely understand the deformation theory of weakly ramified group actions on curves, and its local counterpart. Noteworthy results are the computation of the (local) versal equicharacteristic deformation functor with Kato, the mixed-characteristic functor with Mezard, and the proof of universality for most of those with Byszewski.
With Karim Zahidi, Thanases Pheidas and Shasha Shlapentokh, he has worked on undecidable diophantine problems over the rational numbers. With Zahidi, he proved that the existence of a diophantine model of the integers in the rational numbers defies a conjecture of Mazur, and he found a one-universal-quantifier definition of the integers in the rationals, based on a conjecture about elliptic divisibility sequences. He also studied diophantine storing and other relations between undecidability and elliptic curves.
With Oliver Lorscheid, he studied the theory of toroidal automorphic forms. This is part of a previously dormant approach to the Riemann Hypothesis initiated by Don Zagier in the 1970's. Results include a study of such automorphic forms for function fields of class number one, and some structural results for the space of such forms over number fields, using multiple Dirichlet series.
With Matilde Marcolli, Kamran Reihani and Alina Vdovina, he has worked on the relation between spectral triples (a.k.a. noncommutative Riemannian geometries) and rigidity phenomena for classical spaces such a Riemann surfaces and graphs or buildings.

With Akerblom, Stavenga and van Holten, he studied the construction of explicit solutions to the Jackiw-Pi model on a torus. With Akerblom, he investigated braneworlds with torus structure, and an application of relative entropy in classical gravitational models.
He is currently studying zeta functions in Riemannian and noncommutative geometry, also as a tool in metric Riemannian geometry, relations between anabelian geometry, class field theory and quantum statistical mechanical systems (with Marcolli).