Claire Burrin
Assistant Professor University of Zurich Institute of Mathematics Winterthurerstrasse 190 8057 Zürich, Switzerland Office: Y27K26 Tel.: +41 44 635 58 86 claire.burrin@math.uzh.ch

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My research explores connections between number theory, geometry, and dynamics. I am at the moment particularly interested in counting problems related to spectral geometry and automorphic forms. I am part of the Zurich Dynamics and Ergodic Theory group, and the SwissMAP research network. Before joining UZH, I was a Hill Assistant Professor at Rutgers University, a senior research associate and PhD student at ETH Zurich. My research is currently funded by the Swiss National Science Foundation and the French-Swiss cofunded project Equidistribution in Number Theory. As part of the latter grant, I am currently advertising a 3 years postdoc position, starting Summer 2025; information can be found at Mathjobs.

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Zurich, November 2024

Teaching

In Zurich (2020-)

• 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

• Equidistribution of continuous low-lying pairs of horocycles via Ratner's theorem. Accepted for publication in Comptes Rendus Math. Acad. Sci. Paris (2024).
• On the crossing number of arithmetic curve systems, with Sebastian Baader and Luca Studer. Accepted for publication in Commentarii Math. Helvetici (2024).
• Rational points on ellipsoids and modular forms, with Matthias Gröbner. Preprint (2024).
• 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) (2024), online first.
• Windings of prime geodesics, with Flemming von Essen. Int. Math. Res. Not. (IMRN) 22 (2024), 13931-13963.
• The Manin-Drinfeld theorem and the rationality of Rademacher symbols. Journal de Théorie des Nombres de Bordeaux 34 (3) (2022), 739-753.
• Translates of rational points on expanding horocycles on the modular surface, with Uri Shapira, Shucheng Yu. Mathematische Annalen 382 (2022), 655-717.
• Infinitely many twin prime polynomials of odd degree, with Matthew Isaac. The American Mathematical Monthly 128 (2021), no. 10, 929-935.
• Effective counting for discrete lattice orbits in the plane via Eisenstein series, with Amos Nevo, Rene Rühr, Barak Weiss. L'Enseignement Mathématique 66 (2020), no. 3-4, 259-304.
• Transformation laws for generalized Dedekind sums associated to Fuchsian groups, with Jay Jorgenson, Cormac O'Sullivan, Lejla Smajlović. Acta Arithmetica 196, no.2, 2020.
• Dedekind sums, reciprocity, and non-arithmetic groups. In: Automorphic Forms and Related Topics, Contemporary Mathematics 732, AMS, Providence, RI, 2019.
• Reciprocity of Dedekind sums and the Euler class. Proceedings of the American Mathematical Society 146, no.4, 2018.
• Generalized Dedekind sums and equidistribution mod 1. Journal of Number Theory 172, 2017.
• Dedekind symbols and applications of the spectral theory of Kloosterman sums. Dissertation ETH No. 23634, 2016. ETH medal for outstanding thesis.

Miscellanea

• 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 Research Station, Les Diablerets, Sep 1-5, 2025
• Ergodic Theory and Dynamical Systems Seminar between ETH and UZH (since 2023)
• Ventotene Conference on Groups and Rigidity in the Zimmer Program (GRAZP), September 11-16 2023
• International Seminar on Automorphic Forms (online) (since 2022)
• Swiss Math Society Fall Conference Dynamics and Groups, September 14-16, 2022, at UZH
• Rutgers Number Theory Seminar (2018-2020), Rutgers' Postdoc Colloquium (2017)

Recent & upcoming talks

• Recent developments around joint equidistribution, Hausdorff Institute, Bonn, Mar 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
• Inaugural lecture, UZH (online recording), Dec 2023
• Swiss Math. Society Fall Conference: Swiss Geometry Day, Brig, Oct 2023
• Women in automorphic forms, Konstanz, Oct 2023
• (Minicourse) Geometry and spectra of random hyperbolic surfaces, CRM-ISM Montréal, Jun 2023
• (Minicourse) Elliptics summer school, Physics Department, UZH, Jun 2023:
• Number theory seminar, Shandong University, May 2023:
• Group actions and harmonic analysis in number theory, Oberwolfach, May 2023
• Arithmetic Statistics, Automorphic Forms and Ergodic Methods, MPIM Bonn, Apr 2023
• Algebraic geometry and number theory seminar, Chalmers/Gothenburg, Apr 2023
• Séminaire de géométrie, Bordeaux, Mar 2023

Lectures