**(Function field) arithmetic and automorphic forms** I studied the distribution of *zeros of Eisenstein series for function fields*, with applications to supersingularity of Drinfeld modules.
I gave criteria for the existence of rational *2-power torsion points on Jacobians of hyperelliptic curves over finite fields*.
I studied the ring structure of *rings of Drinfeld modular forms*. With Oliver Lorscheid, I 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 Valentijn Karemaker, I studied the "automorphic version of the anabelian theorems", namely, in how far *Hecke algebras* for general number fields determine the number fields.
With Kato and Janne Kool, I have studied questions of *gonality* from the point of view of *graph theory* (with applications to diophantine problems, such as a lower bound on the gonality of Drinfeld modular curves that is linear in the genus, and to a bound on the degree of modular parametrisations of elliptic curves over function fields). In this connection, I started to study the gonality in certain random graph models.

**Nonarchimedan uniformization** With Fumiharu Kato and Aristides Kontogeorgis, I studied *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.

**Equivariant deformation theory** With Fumiharu Kato, Ariane Mézard and Jakub Byszewski, I computed 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 Mézard, and the proof of universality for most of those with Byszewski.

**Undecidability in number theory** With Karim Zahidi, Thanases Pheidas and Shasha Shlapentokh, I worked on *undecidable diophantine problems over the rational numbers*. With Zahidi, I proved that the existence of a diophantine model of the integers in the rational numbers defies a *conjecture of Mazur*, and found a one-universal-quantifier definition of the integers in the rationals, based on a conjecture about elliptic divisibility sequences. I also studied diophantine storing and other relations between undecidability and *elliptic curves*.
With Jonathan Reynolds, I generalized *divisiblity sequences* to matrices.

**Noncommmutative geometry** With Matilde Marcolli, Kamran Reihani and Alina Vdovina, I have 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; including K-theoretic aspects. With Jan Willem de Jong, I studied the *isospectrality problem* from the (noncommutative) point of view of families of (generalized) zeta functions.
With Matilde Marcolli, I studied relations between *arithmetic geometry, anabelian geometry, class field theory and quantum statistical mechanical systems.* I have also considered this problem for function fields. I am currently also studying *metric geometry of spaces of noncommutative manifolds* (i.e., metrics in the space of spectral triples up to Morita equivalence), and relations between operator K-theory and automorphic forms (with Bram Mesland).

**Mathematical physics** With Nikolas Akerblom, Gerben Stavenga and Jan-Willem van Holten, I studied the construction of explicit solutions to the Jackiw-Pi model on a torus. With Akerblom, I investigated braneworlds with torus structure, and an application of relative entropy in classical gravitational models.