My works on arxiv.

**(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.

**Graphs and gonality** 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 studied the gonality in certain random graph models. I studied graph gonality as a tool in theoretical computer science: with Hans Bodlaender and Marieke van der Wegen, we looked at efficient characterisation of hyperelliptic graphs, and showed that certain classical graph flow problems that are hard when parameterized by treewidth become finite parameter tractable in bounded gonality.

**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. We also developed an analytic counterpart for Mumford curves, and studied automorphism groups of Drinfeld modular curves using non-archimedean uniformization. We described *p*-adic analogues of the Platonic solids.

**Algebraic dynamics in positive characteristic** With Jakub Byszewki and Marc Houben, I studied the *dynamics of endomorphisms of algebraic groups in positive characteristic p*. We consider the rationality/transcendence of the dynamical zeta function and prime orbit distribution in relation to properties of the endomorphism (e.g., its action on the *p*-torsion group scheme). The phenomena we observe in this rigid "algebraic" setup also occur in topological dynamics, e.g. for certain solenoids and cellular automata.

**Undecidability in number theory; divisibility sequences** 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. We also studied pure powers in elliptic divisibility sequences over function fields, using the abc-hypothesis.

**Arithmetic equivalence** With Bart de Smit, Xin Li, Matilde Marcolli and Harry Smit, I proved that isomorphism of global fields can be detected by using L-series (that this is impossible using only zeta functions is the phenomenon of "Arithmetic equivalence"). Following work of Sunada, with Norbert Peyerimhoff, I showed how to detect covering equivalence of certain manifolds using spectra of twisted Laplace operators.

**Automatic series and the Nottingham group** with Jakub Byszewski and Djurre Tijsma, I have shown how to mix the theory of Witt vectors with an effective version of Christol's theorem to construct automata that describe generators of certain finite subgroups of the Nottingham group (the group of formal power series over a finite field without constant term, under substitution); including a theoretical discussion of the sparsity of the corresponding series.

**Mathematical physics and noncommutative geometry ** 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. With Jan-Willem de Jong, 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, graphs or buildings, families of (generalized) zeta functions.