"When the very great physicist Bohr was visiting the great physicist Landau in Moscow he was invited to give a talk to the graduate students with Landau translating. Bohr concluded his talk with the assertion ‘I attribute my success to the fact that I have never been afraid to let my students tell me what a fool I am’. The Russian translation ended ‘I attribute my success to the fact that I have never been afraid to tell my students what fools they are’." (T.W. Korner, In Praise of Lectures)

PhD Students

Oliver Lorscheid, PhD 2008

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 Byszewski, PhD 2009

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 is doing a Habilitation 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 de Jong, PhD 2011

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 Venselaar, PhD 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 Kool, PhD 2013

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.

  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
  5. On the Galois-module structure of polydifferentials of Subrao curves, modular and integral representation theory (with with Fumiharu Kato and Aristides Kontogeorgis). preprint 2013.

Sebastian Klein, PhD 2014

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.

  1. Research Output
  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 Karemaker, PhD 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. Research Output
  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), preprint arXiv:1409.1385 (2014)
  4. Large Galois images for Jacobian varieties of genus 3 curves (with Sara Arias-de-Reyna, Cécile Armana, Marusia Rebolledo, Lara Thomas, Núria Vila), preprint arXiv:1507.05913 (2015), to appear in Acta Arith.
  5. Hecke algebras for GLn over local fields, preprint arXiv:1510.06606 (2015)

Harry J. Smit, PhD 2020

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".

Master Students

Syb(ren) Botma, MSc 2004

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 Bessels, MSc 2005

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 Streng, MSc 2006

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 Teuben, MSc 2006

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

Lotte Weeda (nee van der Zalm), MSc 2008

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 Bod, MSc 2008

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 de Looij, MSc 2009

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 Klein, MSc 2009

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 Matthijssen, MSc 2011

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 Kluck, MSc 2014

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 van Woerden, Msc 2014

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 is pursuing a PhD in algebraic topology in Regensburg.

Peter Lombaers, MSc 2014

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 Hanselman, MSc 2015

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 Smit, Msc 2016

Harry wrote a thesis on class field theory, dynamical systems and L-series. He continued with PhD studies at Utrecht.

Daniel Kroes, MSc 2016

Daniel studied graph reconstruction problems, and extended a method of Myrvold et. al. to certain tridegreed graphs. He went to graduate school at UC San Diego.

Tom van Overbeeke, MSc 2017

Tom is studying the relation between random matrix theory and function field arithmetic.

Lois van der Meijden, MSc 2017

Lois is studying dynamical zeta functions for iterates of rational maps in positive characteristic.

Maxim Faber, MSc 2017

Maxim studied ferromagnets using elliptic functions. Co-supervised with Rembert Duine (ITF).

Marieke van der Wegen, MSc 2017

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

Jelco Bodewes, MSc 2017

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

Bachelor Students

Ryk Westwood, BSc 2002

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

Wouter Waalewijn, BSc 2003

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.

Marius de Leeuw, BSc 2003

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 Kool, BSc 2003

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 Teuben, BSc 2003

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

Willem Maat, BSc 2004

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 de Jong, BSc 2004

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 Streng, BSc 2004

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 Bessels, BSc 2005

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

Joris Borgdorff, BSc 2006

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 and is now doing a PhD in computer science at Amsterdam.

Wouter van der Bilt, BSc 2008

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 Konter, BSc 2009

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.

  1. Research Output
  2. Classifying polygonal algebras by their K_0-group (joint with Alina Vdovina). to appear in Proc. Edinburgh Math.Soc.

Danielle Giovanni, BSc 2009

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 Maes, BSc 2009

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

Maria Velema, BSc 2010

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 Klaasse, BSc 2010

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 Ruiz, BSc 2012

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 Pranger, BSc 2013

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 Staps, BSc 2014

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 Gerken, BSc 2015

Franziska wrote a thesis on the Ihara zeta function of a graph.

Jan Willem van Ittersum, BSc 2015

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. An equivariant version of Lehmer's conjecture on heights. preprint arXiv:1602.01749, to appear in J. Numb. Th.

Jetze Zoethout, BSc 2015

Jetze wrote a thesis about a characterisation of the Bass-Hashimoto edge incidence matrix of a graph.

Thijs van der Gugten, BSc 2015

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.

Alexander Gietelink, BSc 2015

Alexander studied spectral triples on non-commutative analogues of the modular curve.

Lars van de Berg, BSc 2015

Lars wrote a thesis about the proof of the analogue of the Riemann hypothesis for elliptic curves over finite fields.

Ragnar Groot-Koerkamp, BSc 2016

Ragnar constructed an algorithm for the computation of stable gonality of graphs, proving some new theoretical results on the way.

Joost Fransen, BSc 2016

Joost studied generalisations of the casus irreducibilis.

Mees Verheije, BSc 2016

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

Joost Houben, BSc 2016

Joost studied the algorithmic undecidability of the spectral gap problem, following work of Cubitt et. al.



Gunther Cornelissen, Mathematisch Instituut, Universiteit Utrecht

Postal Address:Postbus 80.010, NL-3508TA Utrecht
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E-mail: g.cornelissen@uu.nl