Claire Burrin

Assistant Professor
Institute of Mathematics
University of Zurich
Winterthurerstrasse 190
8057 Zürich, Switzerland

Office: Y27K26
claire.burrin@math.uzh.ch
My research explores connections between geometry, dynamics, and number theory, 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
Zurich, January 2026

Curriculum Vitae

Invited Long Stays at Research Institutes

Grants, Distinctions, Fellowships

Research Group

Postdocs

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Teaching

In Zurich (2020-) At Rutgers (2016-2020)

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).
    Mathematika 72, no. 1, (2026).
    arXiv:2503.08206 [math.DS], DOI [Mat]
    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 363 (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).
    Commentarii Mathematici Helvetici (2025), online first.
    arXiv:2403.05194 [math.GT], DOI [CMH]
    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 Jon Chaika and Samantha Fairchild).
    Journal of the European Mathematical Society (JEMS) (2024), online first.
    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).
    International Mathematics Research Notices (IMRN) 22 (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 34 (2022) no. 3, pp. 739-753.
    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 and Shucheng Yu).
    Mathematische Annalen 382 (2022), pages 655–717.
    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 128(10) (2021), 929–935.
    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, and Barak Weiss).
    L'Enseignement Mathématique 66 (2020), 259-304.
    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

  • A human story of curiosity and relevance, Bulletin of the Swiss Association of University Lecturers (VSH-AEU), special issue on Mathematics (2025)
  • 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

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