Oliver defended his thesis Toroidal automorphic forms for function fields in 2008. In the 1970's, Don Zagier defined a new space of automorphic forms for GL(2), so-called toroidal automorphic forms, defined by the vanishing of their constant Fourier coefficients along all tori. The Fourier coefficient of an Eisenstein series is the zeta function of the quadratic extension that corresponds to the torus, which links toroidal forms to zeros of zeta functions. This thesis is the first published study of this space of toroidal automorphic forms for global function fields. Oliver proves that the space is finite-dimensional. He also shows that the dimension is not twice the genus of the curve (as one would expect from the zeros of the corresponding zeta function), but rather, relates to the class number of the field and the number of cusp forms with vanishing central L-value. He makes a detailed study of the case of elliptic function fields. The tools are a mixture of classical adelic methods and methods more akin to geometric Langlands. On the way, he develops a generalisation of the tree of SL(2) to more general "Graphs of Hecke operators". The research was funded by NWO through my VIDI-project on "Nonarchimedean geometry and automorphic forms". Oliver went on to do a post-doc at MPIM Bonn, became assistant of Markus Reineke in Wuppertal, was Feodor Lynnen Fellow at CUNY and temporary professor at Frankfurt, before joining IMPA (Rio de Janeiro) as tenured researcher.

1. Published results related to PhD research
2. Toroidal automorphic forms for some function fields (with Gunther Cornelissen). Journal of Number Theory 129 (6), 1456-1463, 2009.
3. Toroidal automorphic forms, Waldspurger periods and double Dirichlet series (with Gunther Cornelissen). Multiple Dirichlet series, L-functions and automorphic forms, 131-146, Progr. Math., 300, Birkhauser/Springer, New York, 2012.
4. Automorphic forms for elliptic function fields. Math. Z. 272, no. 3-4, 885-911, 2012.
5. Graphs of Hecke operators. Algebra Number Theory 7, no. 1, 19-61, 2013.
6. Toroidal automorphic forms for function fields. Israel J. Math. 194, no. 2, 555-596, 2013.

Jakub defended his thesis Cohomological aspects of equivariant deformation theory, co-supervised by Ariane Mezard (Jussieu), in 2009. In his thesis, he accomplishes three things: he provides an elementary example of a linear representation of a finite group whose universal deformation space is not a complete intersection; he classifies exactly which deformation functors of actions of finite groups as automorphisms of formal power series are pro-representable (when the ramification is weak); and he develops a general theory of devissage for deformation functors of the above type. On the way, he presents a new pro-representability criterion, and various results on the Nottingham group. The research was funded by NWO through my VIDI-project on "Nonarchimedean geometry and automorphic forms" and a Van Gogh grant with France. Jakub has a tenured position at the Mathematical Institute of Jagiellonian University in Cracow.

1. Published results related to PhD research
2. A universal deformation ring which is not a complete intersection ring. C. R. Acad. Sci. Paris, Ser. I 343, Issue 9, 565–568, 2006.
3. Which weakly ramified group actions admit a universal formal deformation? (with Gunther Cornelissen). Ann. Inst. Fourier 59, no. 3, pp. 877-902, 2009.
4. Deformation functors of local actions. arxiv:1112.0352 (23pp.), 2011.
5. Un anneau de deformation universel en conducteur superieur (with Gunther Cornelissen and Fumiharu Kato). Proc. Japan Acad. Sci., Ser. A 88, no. 2, 25-27, 2012.

Jan Willem defended his thesis Zeta function rigidity - a view from noncommutative geometry in 2011. In his thesis, Jan Willem first shows how to construct a finitely summable spectral triple for a graph on the Cantor set whose family of zeta functions reconstructs the graph.Then he remedies the isospectrality problem for closed smooth Riemannian manifolds by showing that a given diffeomorphism of Riemannian manifolds is an isometry precisely if a family of spectral zeta functions indexed by the algebra of smooth functions match by pullback via the diffeomorphism. In the final chapter, this is used to define a metric in the space of Riemannian manifolds, based on a notion of length of a diffeomorphism, that measure the distance between zeta function. The research was funded by NWO through my VICI-project on "From Arithmetic Geometry to Noncommutative Riemannian Geometry, and back". Jan Willem is researcher at UL Transaction Security.

1. Published results related to PhD research
2. Graphs, spectral triples and Dirac zeta functions. P-Adic Numbers, Ultrametric Analysis, and Applications 1, 286-296, 2009.
3. The spectral length of a map between Riemannian manifolds (with Gunther Cornelissen). J. Noncommut. Geom. 45, 721-758, 2012.

Jan Jitse defended his thesis Classification and equivalences of noncommutative tori and quantum lens spaces in 2012. In his thesis, he gives a classification of structures of spectral triple on equivariant noncommutative tori in arbitrary dimension. This was done by a very different method by Paschke and Sitarz in dimension 2; Jan Jitse uses Connes reconstruction theorem. An amusing corollary is a criterion when a set of matrices spans a Clifford algebra, for which a direct linear algebra proof is still not know. Then he proves that Morita equivalences of such noncommutative spin n-tori is symmetric (Morita equivalence of spectral triples is in general not symmetric, and this provides the first non-trivial example). Finally, he defined and studied noncommutative lens spaces and studied their equivalences. The research was funded by NWO through my VICI-project on "From Arithmetic Geometry to Noncommutative Riemannian Geometry, and back". Jan Jitse went to Caltech for a post-doc, and then to Göttingen, and is now a Research Engineer at ASML.

1. Published results related to PhD research
2. Classification of spin structures on the noncommutative n-torus.Journal of Noncommutative Geometry 7, 2013.
3. Morita “equivalences” of equivariant torus spectral triples.Letters in Mathematical Physics 103, 2013.
4. The Geometry of Quantum Lens Spaces: Real Spectral Triples and Bundle Structure (with Andrzej Sitarz), Mathematical Physics, Analysis and Geometry 18:9, December 2015.

Janne defended her thesis Graphs, Curves and Dynamics in 2013. In her thesis, she first presents a rigidity criterion for Mumford curves ("p-adic Riemann surfaces") which looks a bit like Mostow rigidity, but, in addition to absolute continuity of a boundary measure, involves preservation of finitely many algebraic relations between measures that arise from harmonic cocycles as in the work of Teitelbaum. In her thesis, she also introduced dynamical systems in which the measure is p-adic valued, and made the first study of basic properties of recurrence and entropy for such systems (which are rather different from real-valued measurable dynamics). Then she introduces a notion of "stable gonality" of a graph and proves the analogue of a famous theorem of Li and Yau from differential geometry about gonality of Riemann surfaces for the graph case (the strictly analogous statement is false and has to be altered, and the proof is a mixture of probability on graphs and re-engineering graphs with bounded change of invariants). Then she proves that the gonality of a non-archimedean curve is bounded below by that of its stable reduction graph. The results can be combined with the theory of automorphic forms to prove a lower bound on the gonality of modular curves that is linear in the genus (analogue of a result of Abramovich) and to prove a conjecture of Papikian on the degree of the modular parametrisation of elliptic curves over function fields. The research was funded by NWO through my VICI-project on "From Arithmetic Geometry to Noncommutative Riemannian Geometry, and back". Janne went to MPIM Bonn for a post-doc, and then to the "Mathematics of Reaction Networks" group at Copenhagen, and is now a researcher at WUR (Wageningen University and Research).

1. Published results related to PhD research
2. Measure theoretic rigidity for Mumford Curves (with Gunther Cornelissen). Ergodic Theory and Dynamical Systems 33, 851-869, 2013.
3. Dynamics measured in a non-Archimedean field. P-Adic Numbers Ultrametric Anal. Appl. 5, 1-13, 2013.
4. A combinatorial Li-Yau inequality and rational points on curves (with Gunther Cornelissen and Fumiharu Kato), Math. Ann. 361:1 211-258, 2015

Sebastian defended his thesis Chow groups and intersection products for tensor triangulated categories, co-supervised with Paul Balmer (UCLA), in 2014. In his thesis, Sebastian studies the definition of Chow group of a tensor triangulated category, following Balmer. He proves that this gives the usual Chow group for regular schemes, he establishes the functoriality of the construction, shows how to compute the Chow group of the stable category of modular representations of cyclic p-groups and of the Klein four group in characteristic two. He also generalizes the definition to the relative setting, where one tensor-triangulated category acts on another one as in the work of Greg Stevenson. In the final part of his thesis, he extends the concept of Frobenius pair to that of a tensor Frobenius pair, establishes all standard properties, and uses this to define an alternative Chow group, on which it is then possible to define an intersection product through the use of K-theory, under the assumption of a tensor-triangulated version of the Gersten conjecture. The research was partially funded by NWO through my VICI-project on "From Arithmetic Geometry to Noncommutative Riemannian Geometry, and back", and partially through the Faculty Focus Area on "Foundations of Natural Sciences". He continued with a post-doc at Antwerp and now works for Jibes IT Consultancy.

1. Published results related to PhD research
2. Chow groups of tensor triangulated categories, Journal of Pure and Applied Algebra 220:4 1343-1381, 2016.
3. Intersection products for tensor triangular Chow groups, Journal of Algebra 449, 497-538, 2016.

Valentijn defended her thesis Hecke algebras, Galois representations, and abelian varieties in 2016 and specializes in number theory and arithmetic geometry. She studied the inverse Galois theory problem for symplectic groups, and an automorphic version of the anabelian theorem of Neukirch, in which the absolute Galois group is replaced by a Hecke algebra. The research is funded by NWO through a grant from the GQT-cluster. She continued with a three year post-doc at UPenn.

1. Published results related to PhD research
2. Galois representations and Galois groups over Q (with Sara Arias-de-Reyna, Cécile Armana, Marusia Rebolledo, Lara Thomas, Núria Vila), Proceedings of Women in Numbers Europe - Research Directions in Number Theory.
3. Hecke algebra isomorphisms and adelic points on algebraic groups (with Gunther Cornelissen), Doc. Math. 22 851-871 (2017).
4. Large Galois images for Jacobian varieties of genus 3 curves (with Sara Arias-de-Reyna, Cécile Armana, Marusia Rebolledo, Lara Thomas, Núria Vila), Acta Arith. 174, 339-366 (2016).
5. Hecke algebras for GLn over local fields, Arch. Math. 107, 341-353 (2016).
6. Fully maximal and fully minimal abelian varieties (with R. Pries), arxiv, to appear in JPAA.

Harry started his PhD research in 2016 and specializes in number theory and arithmetic geometry. He will study higher dimensional class field theory and L-series for curves over finite fields. The research is funded by NWO through a block grant for the graduate programme "Utrecht Geometry Centre".

1. Published results related to PhD research
2. Reconstructing global fields from dynamics in the abelianized Galois group (with Gunther Cornelissen, Matilde Marcolli and Xin Li), arxiv
3. Reconstructing global fields from Dirichlet L-series (with Gunther Cornelissen, Bart de Smit, Matilde Marcolli and Xin Li)arxiv

Jan-Willem started his PhD research in 2017 and specializes in number theory, modular forms combinatorics. He studies the Block-Okounkov formula and is cosupervised by Don Zagier (MPIM Bonn/ICTP) The research is funded by NWO through a block grant for the graduate programme "Utrecht Geometry Centre".

Marieke started her PhD research in 2017 and specializes in graph algorithms, especially in relation to graph divisors and gonality. She is cosupervised by Hans Bodlaender (Utrecht CS) and the research is funded by a collaborative grant from the Utrecht CS department.

1. Published results related to PhD research
2. Recognizing hyperelliptic graphs in polynomial time (with Jelco Bodewes, Hand Bodlaender and Gunther Cornelissen), arxiv, extended abstract in: Proceedings of the 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2018), Springer LNCS vol. 11159, pp. 52-64 (2018) [Best Student Paper Award].
3. Stable gonality is computable (with Ragnar Groot Koerkamp), arxiv

Timo works for Booking.com and is doing his PhD in part-time, working on mathematical physics (currently, quantum field theory) in relation to algebra algorithms, co-supervised by Ana Ros Camacho.

Syb wrote a thesis on Tro-p-adical geometry, co-supervised by Jan Stienstra, in which he outlined the theory of amoeba of p-adic curves and searched for an axiomatisation of the planar geometry of tropical p-adic lines and conics.

Sander wrote a thesis called One step beyond the solvable equation, in which he collected literature about the solution of the fifth degree equation using elliptic curves, from historical sources to the PhD thesis of Edray Goins. Sander became a math highschool teacher and then switched to IT.

Marco wrote a thesis on Elliptic divisibility sequences with complex multiplication, in which he developed the theory of elliptic divisibility sequences in which the indices are not integers, but general endomorphisms of an elliptic curve. In case of complex multiplication, one gets a sequence that interpolates the usual elliptic divisibility sequence. Marco proves the existence of primitive divisors (not using the standard estimates, which don't suffice, but rather more detailed estimates of Sinou David). Marco went on to do a PhD in Leiden in computational number theory, and after some time in Warwick, is now assistant professor in Leiden.

1. Research Output
2. Divisibility sequences for elliptic curves with complex multiplication. Algebra & Number Theory 2, 183-208 (2008).

Ruden wrote a thesis on The Riemann-Roch theorem for number fields, doing "Arakelov geometry in dimension zero".

Lotte wrote a thesis on Arithmetic equivalence, in which she studied when number fields and function fields have the same zeta function, including some new work on equality of the Goss zeta functions of two function fields. Lotte went to work for ORTEC.

1. Research Output
2. Arithmetic equivalence for function fields, the Goss zeta function and a generalisation (with Gunther Cornelissen and Aristides Kontogeorgis). Journal of Number Theory 130 (4), 1000-1012 (2010).

Esther wrote a thesis on Hilbert's tenth problem, with very streamlined proofs of the main results. She went on to do a PhD at Utrecht on monodromy of hypergeometric function and now works for Gridline.

Rutger wrote a thesis on Primes in elliptic divisbility sequences, in which he reformulates in modern language a theorem of Morgan Ward on periodicity of elliptic divisibility sequences, and then studies experimentally the periodicity with different sign choices. Rutger became a teacher of mathematics and Latin.

Sebastian wrote a thesis on Reconstructive geometry in certain triangulated categories, in which he proves the reconstruction theorem of Bondal and Orlov, and then uses this to define cycles and equivalences of cycles entirely categorically. He went on to do his PhD thesis at Utrecht on this same topic from a much more advanced point of view. His master thesis won the 2010 GQT prize for best Dutch master thesis in Geometry and Quantum Theory.

Jori wrote a thesis on Common Divisors of Elliptic Divisibility Sequences over Function Fields, in which he fleshed out a paper by Silverman on that topic, including correcting and complementing some of the proofs, and doing rather largish computations in search for further good conjectures. Jori became a quant in Amsterdam.

Florian wrote a thesis on A metric in the space of spectral triples, in which he surveyed the theory of Hilbert C*-modules and Mesland's theory of correspondences, and then introduces the length of such a correspondence (following Cornelissen and Mesland) and computes it is case of a scaling correspondence between circles the quotient of whose radii is an integer. Florian now leads a science remedial teaching institute.

Koen wrote a thesis on The Point Counting Function on Elliptic Curves, following a recent book by Jean-Pierre Serre on a similar function for an arbitrary curve. He makes the theory very explicit and replaces etale cohomology by the Tate module, and uses division polynomials, he computes various examples, and presents a complete proof of Chebotarev's density theorem. Koen did a PhD in algebraic topology in Regensburg and now works in consultancy in Germany.

Peter wrote a thesis on Dessins d’Enfants for surfaces, in which he surveys Belyi's theorem and the theory of dessins d'enfants, after which he describes the topology of the moduli spaces of curves of genus zero with 4 and 5 marked points, and defines the analogue of dessins for surfaces instead of curves. Peter is doing a PhD in Porto.

Jeroen wrote a thesis on Semi-stable reduction of curves and morphisms of curves, in which he studies a degree bound on the minimal field required for the extension of a morphism of curves to semistable models. He continued to pursue a PhD in (algorithmic) arithmetic geometry at Ulm.

Harry wrote a thesis on Global field isomorphisms: a class field theory approach. He continued with PhD studies at Utrecht.

Daniel studied graph reconstruction problems, and extended a method of Myrvold et. al. to certain tridegreed graphs in his thesis The edge reconstruction conjecture for graphs. He went to graduate school at UC San Diego.

Tom studied the relation between random matrix theory and function field arithmetic in his thesis The Euler totient function in short intervals.

1. Research Output
2. The variance of the Euler totient function. arxiv

Maxim studied ferromagnets using elliptic functions. He was co-supervised with Rembert Duine (ITF). We became an Intern at ING Bank.

Marieke studied computational aspects of stable gonality of graphs, in particular, recognition of hyperelliptic graphs. Co-supervised with Hans Bodlaender (CS).

1. Research Output
2. Recognizing hyperelliptic graphs in polynomial time (with Jelco Bodewes, Hand Bodlaender and Gunther Cornelissen), arxiv

Marieke studied computational aspects of divisorial gonality of graphs, in particular, recognition of hyperelliptic graphs. Co-supervised with Hans Bodlaender (CS). She went on to do a PhD at Utrecht.

1. Research Output
2. Recognizing hyperelliptic graphs in polynomial time (with Hand Bodlaender, Gunther Cornelissen and Marieke van der Wegen), arxiv

Sophie was supervised by Daniel Dadush (CWI) and wrote a thesis about the simplex method. She went on to do a PhD with Daniel.

1. Research Output
2. A Friendly Smoothed Analysis of the Simplex Method (with Daniel Dadush) arxiv, STOC 2018.

Lois studied dynamical zeta functions for iterates of rational maps in positive characteristic. She went to work at an investment bank.

Djurre studied the construction of power series of finite order using automata theory, discovering some new examples. He went on to do a PhD in Dusseldorf.

Marc studied dynamically affine maps and endomorphisms of algebraic groups in positive characteristic. He went on to do a PhD at Leuven.

Jeroen studied the pre-representability of deformation functors of group actions on curves. He went on to do a PhD at Amsterdam.

Joost is studying essential dimension.

Ryk wrote a thesis on Dirichlet's theorem for polynomials, including his own study of constants in the asymptotics.

Wouter wrote a thesis on Change of the rank of elliptic curves under field extensions . Wouter continued with a masters in physics and did his PhD in physics at MIT. He is currently assistant professor of physics at Amsterdam.

Wouter wrote a thesis on integral points on elliptic curves, fleshing out the method of elliptic logarithms. In particular, he solves the problem "for which integers m and n is the sum of the first m integers equal to the sum of the first n squares?". He continued with a combined masters in mathematics and physics and did his PhD in physics at Utrecht, after which he went to MPI Gravitationsphysik and ETHZ.

Martijn wrote a thesis on The local-to-global principle for conics and elliptic curves. He continued with a combined masters in mathematics and physics and did his PhD in mathematics at Oxford. He worked at Imperial College and Vancouver and is currently assistant professor at Utrecht.

Ruden wrote a thesis on The last theorem of Fermat for regular primes. He continued with a masters at Utrecht.

Willem wrote a thesis on Collatz Problems, in which he presented his own interpretation and generalization of the "3n+1"-problem of Collatz. He continued with a masters at Utrecht.

Jan Willem wrote a thesis on Cyclotomic extensions of Q and F_p(t), including the theory of the Carlitz module. He continued with a masters at Utrecht.

Marco wrote a thesis on Analytic proofs of quadratic and quartic reciprocity by Eisenstein, as presented in the book by Lemmermeyer. He continued with a masters at Utrecht.

Sander wrote a thesis on Primes of the form x^2+ny^2. He continued with a masters at Utrecht.

Joris wrote a thesis on Expanding graphs, including a computer program for the computation of Cheeger constants. After a year in South Africa working in health care IT, he continued with a masters in computer science, did a PhD in computational science at Amsterdam, and is now Data Engineer.

Wouter wrote a thesis on Elliptic curves and class numbers, following a forgotten paper by Duncan Buell that Serge Lang pointed out to us. He continued with a masters at Utrecht.

Johan wrote a thesis on K-theory of graphs and buildings, based on joint research with Alina Vdovina (Newcastle) that he visited with a grant from the Monna Fund. He continued with a masters at Utrecht, and then went to graduate school at Northwestern to do a PhD in algebraic topology.

1. Research Output
2. Classifying polygonal algebras by their K_0-group (joint with Alina Vdovina). Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 485-497

Danielle wrote a thesis on Elementary proofs of Dirichlet's theorem for polynomials, transporting some elementary proofs of Dirichlet's theorem on primes in arithmetic progressions using cyclotomic polynomials to rational function fields.

Rianne wrote a thesis on Cryptography using elliptic curves. She went on to do a masters at Utrecht.

Maria wrote a thesis on Graph puzzles and exceptional geometries, following a paper by Wilson. She went on to do a masters in logic at Amsterdam.

Thom wrote a thesis on Distances between metric spaces, trying to squeeze the Gromov-Hausdorff distance between flat tori between computable bounds. He went on to do a masters at Utrecht.

Ederick wrote a thesis on Finite p-groups are nilpotent, proving precisely that (and showing a lot of examples). He went on to do a masters in Stochastics and Financial Mathematics.

Willem wrote a thesis on Riemann's explicit formula for the prime counting function, in which he writes down all details in modern language of Riemann's famous proof. He went on to do a masters at Utrecht.

Merlijn wrote a thesis on Sum and Difference Sets, in which he develops some additive combinatorics and studies the when the inequalities in the bounds "square root of the doubling constant is less than or equal to the difference constant, is less than or equal to the square of the doubling constant" become equalities.

1. Research Output
2. The relative size of sumsets and difference sets. Integers A42, 2015.

Franziska wrote a thesis on the Ihara zeta function of a graph, and continued with a masters in Mathematics in Muenster (D).

Jan Willem wrote a thesis on Mahler measure and Möbius transformations. He generalizes results of Zhang, Zagier and Dresden (for certain Möbius transformations of degree 2 and 3) to arbitrary finite groups of Möbius transformations, giving, under certain conditions, a lower bound for the sum of the logarithmic Mahler measures of the orbit of a polynomial under such a group.

1. Research Output
2. A group-invariant version of Lehmer's conjecture on heights. J. Number Theory 171 (2017), 145-154.

Jetze wrote a thesis about a characterisation of the Bass-Hashimoto edge incidence matrix of a graph, continued with a masters in mathematics and in philosophy at Utrecht and is now doing a PhD in mathematicsl logic at Utrecht.

Thijs wrote a thesis about regular prime numbers (including Kummmer's proof of Fermat for regular primes). He then transferred the definition to Carlitz-Bernoulli-numbers for the ring of polynomials over a finite field, and proved that in that case, there are infinitely many regular prime polynomials; he continued with a masters at Utrecht.

Alexander studied spectral triples on non-commutative analogues of the modular curve. He continued with a masters at Utrecht and Paris.

Lars wrote a thesis about the proof of the analogue of the Riemann hypothesis for elliptic curves over finite fields. He continued with a masters at Utrecht.

Ragnar constructed an algorithm for the computation of stable gonality of graphs, proving some new theoretical results on the way. He continued with a masters in computer science and mathematics at Oxford and now works at Google Research Zurich.

1. Research Output
2. Stable gonality is computable (with Marieke van der Wegen), arxiv

Joost studied generalisations of the casus irreducibilis, and continued with a masters at Utrecht.

Mees studied van der Waerden's paper on the probability that an integral polynomial has maximal Galois group.

Joost studied the algorithmic undecidability of the spectral gap problem, following work of Cubitt et. al. He continued with a masters in Oxford and now works for Microsoft Research in Seattle.

Eva studied the Inverse Galois problem using Galois representations associated to Drinfeld modules. She continued with a masters at Utrecht.

Rens studied a polynomial time algorithm for deciding whether or not a given polynomial is solvable.

Laurent studied notions of provable security in cryptography.

Alex studied the prime number theorem in polyomial rings. .