Claire Burrin

Assistant Professor Institute of Mathematics University of Zurich Winterthurerstrasse 190 8057 Zürich, Switzerland Office: Y27K26 claire.burrin@math.uzh.ch

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My research explores connections between number theory, geometry, ergodic theory and dynamics, with an emphasis on the question: How do arithmetic and geometric structures give rise to random-like phenomena, and what analytic tools allow us to describe them? I am part of the Zurich Dynamics and Ergodic Theory group; the SwissMAP research network; the SNSF-ANR joint project Equidistribution in Number Theory, with Menny Aka, Régis de la Bretèche, Farrell Brumley, Sary Drappeau, Manfred Einsiedler, Florent Jouve, Emmanuel Kowalski, Philippe Michel, Nicolas de Saxcé.

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Zurich, September 2025

Curriculum Vitae

• Since 2022: Assistant Professor, University of Zurich
• 2020-2022: Senior Research Associate (Oberassistentin), ETH Zurich
• 2016-2020: Hill Assistant Professor, Rutgers the State University of New Jersey
  Postdoc mentors: Henryk Iwaniec, Alex Kontorovich


• 2012-2016: PhD, ETH Zurich; Visiting Student, Princeton University (2013)
• 2007-2012: BSc/MSc, ETH Zurich; M1, Ecole Polytechnique
  PhD/MSc advisor: Alessandra Iozzi

Visiting Positions

Mittag-Leffler Institute (Member, 2025); Hausdorff Research Institute (Junior Fellow, 2021); Centre de Recherche Mathématique (virtual member, 2020); MSRI (Program Associate, 2015)

Grants, Distinctions

• 2025-2029: SNSF Lead Agency Project Grant Equidistribution in Number Theory
• 2022-2027: SNSF PRIMA Career Grant Spectral Techniques in Dynamics and Number Theory
• 2020: CRM-ISM Postdoctoral Fellowship (didn't go through because of the pandemic)
• 2017-2019: AMS/Simons Foundation Travel Grant
• 2017: ETH Medal for outstanding PhD thesis and money-prize

Research Group

Postdocs

• Vikram Giri (SNSF Ambizione Fellow) 2025-2029
• Tariq Osman 2025-2028

PhD Students

• Elias Dubno 2022-2026
  • A central limit theorem for the winding of low-lying closed geodesics. Preprint, 2025.
    arXiv [math.NT].
    Abstract: We show that the winding number of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by essentially any notion of length.
  • Zaremba's conjecture for geometric sequences: an algorithm. The American Mathematical Monthly, 2024.
    arXiv [math.NT], DOI [AMM].
    Abstract: There are only a handful of explicit sequences known to satisfy the strong version of Zaremba's conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba's conjecture.

Teaching

In Zurich (2020-)

• Fall 2025: Algebra (UZH)
• Fall 2024: Elementary number theory (UZH)
• Spring 2024: Complex analysis (UZH)
• Spring 2023: Complex analysis (UZH)
• Fall 2022: Spectral theory of hyperbolic surfaces (CUSO Graduate Lectures, Swiss Doctoral Program)
• Spring 2022: Topics in number theory: L-functions and modular forms (graduate course, UZH)
• Spring 2021: Spectral theory of hyperbolic surfaces (graduate course, ETH)

At Rutgers (2016-2020)

• Spring 2020: Probability theory, Honors linear algebra
• Fall 2019: Probability theory
• Spring 2019: Number theory
• Fall 2018: Abstract algebra, Honors linear algebra
• Spring 2018: Abstract algebra
• Fall 2017: Geometry
• Spring 2017: Linear algebra
• Fall 2016: Linear algebra

Publications and Preprints

• Modular systoles are extremal for the crossing number, with Hugo Parlier.
  arXiv:2508.20958 [math.GT].
  Abstract: Taken as a family of curves on a family of surfaces, we show that the growth rate of crossing numbers for systoles of congruence surfaces is optimally small among all sets of curves of the same cardinality lying on the same topological surface.


• The Brjuno and Wilton functions, with Seul Bee Lee and Stefano Marmi.
  arXiv:2503.08206 [math.DS].
  Abstract: We show that the Brjuno function from one-dimensional holomorphic dynamics and the Wilton function from analytic number theory realize two facets of the same function, and explore these relations based on numerical observations.


• Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere, with Matthias Gröbner.
  arXiv:2502.17678 [math.NT].
  Abstract: We investigate the equidistribution properties of rational points on the 2-sphere at small scales, and discuss connections to covering radius, intrinsic Diophantine approximation, and a conjecture of Linnik.


• Equidistribution of continuous low-lying pairs of horocycles via Ratner's theorem. Comptes Rendus Mathématique, 2025.
  arXiv:2408.03348 [math.DS], DOI [CRM].
  Abstract: We record an alternative proof of a recent joint equidistribution result of Blomer and Michel, based on Ratner's topological rigidity theorem.


• On the crossing number of arithmetic curve systems, with Sebastian Baader and Luca Studer. Accepted for publication in Commentarii Mathematici Helvetici.
  arXiv:2403.05194 [math.GT].
  Abstract: We show that the family of systoles of hyperbolic surfaces associated with congruence lattices have asymptotically minimal crossing number.


• Rational points on ellipsoids and modular forms, with Matthias Gröbner.
  arXiv:2403.05194 [math.NT].
  Abstract: We establish new bounds towards the counting and equidistribution of rational points on spheres and other ellipsoids, in what may be viewed as a textbook display of the 'unreasonable effectiveness' of modular forms.


• Pairs in discrete lattice orbits with applications to Veech surfaces, with Samantha Fairchild and an appendix by Jon Chaika. Journal of the European Mathematical Society (JEMS), online first 2024.
  arXiv:2211.14621 [math.DS], DOI [JEMS].
  Abstract: We prove Siegel-Veech and Rogers-type formulas for discrete lattice orbits in the plane from which we derive new results for saddle connections on Veech surfaces. This includes an estimate used in the appendix to prove that pairs of translations flows on a Veech surface are disjoint for almost every set of directions.


• Windings of prime geodesics, with Flemming von Essen. Int. Math. Res. Not. (IMRN), 2024.
  arXiv:2209.06233 [math.NT], DOI [IMRN].
  Abstract: We construct a winding number for closed geodesics on any cusped hyperbolic surface. For various arithmetic families of surfaces, access to the spectral theory of automorphic forms yields statistical results on the distribution of closed geodesics with respect to their winding.


• The Manin-Drinfeld theorem and the rationality of Rademacher symbols. Journal de Théorie des Nombres de Bordeaux, 2022.
  arXiv:2012.01147 [math.NT], DOI [JTNB].
  Abstract: For any noncocompact Fuchsian group, we show that periods of the canonical differential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols. This result provides a relation between Rademacher symbols and the Manin-Drinfeld theorem, including a new proof -- and extensions -- of the latter.


• Translates of rational points on expanding horocycles on the modular surface, with Uri Shapira, Shucheng Yu. Mathematische Annalen, 2022.
  arXiv:2009.13608 [math.DS], DOI [MathAnn].
  Abstract: We study the sparse equidistribution problem for (translates of) rational points along a sequence of expanding closed horocycles on the modular surface.


• Infinitely many twin prime polynomials of odd degree, with Matthew Isaac. The American Mathematical Monthly, 2021.
  arXiv:2101.08038 [math.NT], DOI [AMM].
  Abstract: We discuss the construction of a new infinite family of twin prime polynomials and relations to Wieferich primes and combinatorial Bell numbers.


• Effective counting for discrete lattice orbits in the plane via Eisenstein series, with Amos Nevo, Rene Rühr, Barak Weiss. L'Enseignement Mathématique, 2020.
  arXiv:1905.01493 [math.DS], DOI [LEM].
  Abstract: We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R) acting linearly on the plane. Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.


• Transformation laws for generalized Dedekind sums associated to Fuchsian groups, with Jay Jorgenson, Cormac O'Sullivan, Lejla Smajlović. Acta Arithmetica 196, no.2, 2020.
  arXiv:1904.11333 [math.NT], DOI [AA].
  Abstract: We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants.


• Dedekind sums, reciprocity, and non-arithmetic groups. In: Automorphic Forms and Related Topics, Contemporary Mathematics 732, AMS, Providence, RI, 2019.
  arXiv:1809.03038 [math.NT], DOI [CONM].
  Abstract: This note surveys recent developments of some aspects of Dedekind sums for (non-uniform) lattices in SL(2,R) otherwise referred to as Dedekind symbols.


• Reciprocity of Dedekind sums and the Euler class. Proceedings of the American Mathematical Society 146, 2018.
  arXiv:1609.03588 [math.NT], DOI [PAMS].
  Abstract: We prove the reciprocity law for Dedekind symbols, based on a concrete realization of the Euler class. As application, we obtain a formula for Dedekind sums for Hecke triangle groups in terms of continued fractions.


• Generalized Dedekind sums and equidistribution mod 1. Journal of Number Theory 172, 2017.
  arXiv:1509.04429 [math.NT], DOI [JNT].
  Abstract: Based on their relation to modular forms, Dedekind sums may be viewed as functions on the cusp set of SL(2,Z). We introduce a compatible notion, which we name Dedekind symbols, for any non-cocompact lattice in SL(2,R), and prove an equidistribution mod 1 result for its values.


• Dedekind symbols and some applications of the spectral theory of Kloosterman sums. Dissertation ETH No. 23634, 2016. ETH medal for outstanding thesis. DOI

Miscellanea

• Sphere packing and magic functions, slides for an undergraduate talk as part of the May12 initiative (2024)
• Pairs of saddle connections, in: Group actions and harmonic analysis in number theory, Oberwolfach report No. 22/2023
• Minicourse notes: Counting, equidistribution, and spectral gap - Ventotene Conference on Counting Problems (2021)
• A dynamical outlook on orbital counting problems (with Xin Zhang) (2014)
• Quasihomomorphisms and refinements of property (T) - Master thesis, ETH Zürich, 2012

Conferences and seminars

I am/was a co-organiser of the following events

• Ventotene Conference on higher dimensional hyperbolicity, Sep 8-12, 2025
• International Workshop on Automorphic Forms, SwissMAP Station, Les Diablerets, Sep 1-5, 2025
• Equidistribution in Number Theory Hybrid Seminar Series (since 2025)
• Ergodic Theory and Dynamical Systems Seminar joint seminar ETH-UZH (since 2023)
• Ventotene Conference Groups and Rigidity in the Zimmer Program, Sep 11-16 2023
• International Seminar on Automorphic Forms (since 2022)
• Swiss Math Society Fall Conference Dynamics and Groups, Sep 14-16, 2022, at UZH
• Rutgers Number Theory Seminar (2018-2020)
• Rutgers Postdoc Colloquium (2017, co-founder)

Recent & upcoming talks

• Workshop Group Actions and Harmonic Analysis in Number Theory, Oberwolfach, May 2026
• Conference Ergodic theory, fractal geometry, Diophantine approximation, Warsaw, May 2026
• Workshop Illustrating number theory, IHP, Paris, Mar 2026
• Workshop Explorations in Geometry, Fribourg, Nov 2025
• First Meeting Equidistribution in Number Theory, Paris, Jun 2025
• LG&TBQ2: Geometry, Topology, and Dynamics, CRM Montreal, Jun 2025
• Combinatorics, arithmetic and geometry working group meeting, Paris, Jan 2025
• Colloquium, University of Bern, Oct 2024
• Integral structures in geometry and representation theory, session on Geometry and analysis on locally symmetric spaces, Paderborn, Sep 2024
• Developments in modern mathematics, Goettingen, Sep 2024
• Combinatorial geometry and number theory, EPFL, Aug 2024
• Workshop on Dynamical Systems at ICTP, Trieste, Aug 2024
• International Conference on L-Functions and Automorphic Forms, Vanderbilt, May 2024
• Sphere packings and magic functions, a talk part of the May12 initiative, May 2024 (slides)
• Workshop in analytic number theory, Mittag-Leffler, Mar 2024
• Number theory seminar, University of Virginia, Feb 2024
• Séminaire de théorie ergodique de Rennes, Jan 2024

Lectures