Hannah K. Larson

I am an Assistant Professor of Mathematics at the University of California, Berkeley. I am also a Clay Research Fellow for 2022-2027. I was awarded a 2024 Maryam Mirzakhani New Frontiers Prize. A short article I wrote about my research was featured in Berkeley's Fall 2023 Department Newsletter (pages 6-7).

I received my PhD from Stanford University in June 2022. My thesis, Brill-Noether theory over the Hurwitz space, was advised by Ravi Vakil and won the Hertz Thesis Prize. During the year 2022-2023, I was a Junior Fellow at the Harvard Society of Fellows.

My email address is hlarson at berkeley dot edu. Here is my CV.

Papers and Preprints:

• Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces [joint with Sameera Vemulapalli].
• The Chow ring of the universal Picard stack over the hyperelliptic locus.
• Maximal Brill--Noether loci via the gonality stratification [joint with Asher Auel and Richard Haburcak].
• Extensions of tautological rings and motivic structures in the cohomology of Mbar_{g,n} (Forum of Math, Pi) [joint with Samir Canning and Sam Payne].
• The embedding theorem in Hurwitz--Brill--Noether theory [joint with Kaelin Cook--Powell, Dave Jensen, Eric Larson, and Isabel Vogt].
• The bielliptic locus in genus 11 (Michigan Math Journal) [joint with Samir Canning].
• The eleventh cohomology group of Mbar_{g,n} (Forum of Math, Sigma) [joint with Samir Canning and Sam Payne].
• On the Chow and cohomology rings of moduli spaces of stable curves (Journal of the European Math Society) [joint with Samir Canning].
• The rational Chow rings of moduli spaces of hyperelliptic curves with marked points (Annali della Scuola Normale Superiore) [joint with Samir Canning].
• The integral Picard groups of low-degree Hurwitz spaces (Math Zeitschrift) [joint with Samir Canning].
• Chow rings of low-degree Hurwitz spaces (Crelle) [joint with Samir Canning].
• On an equivalence of divisors on M_{0,n} bar from Gromov-Witten theory and conformal blocks (Transformation Groups) [joint with Linda Chen, Angela Gibney, Lauren Cranton Heller, Elana Kalashnikov, and Weihong Xu].
• The intersection theory of the moduli space of vector bundles on P1 (Canadian Math Bulletin).
• The Chow rings of the moduli spaces of curves of genus 7, 8, and 9 (Journal of Algebraic Geometry) [joint with Samir Canning].
• Tautological classes on on low-degree Hurwitz spaces (International Math Research Notices) [joint with Samir Canning].
• Global Brill-Noether theory over the Hurwitz space (Geometry and Topology) [joint with Eric Larson and Isabel Vogt].
• Refined Brill-Noether theory for all trigonal curves (European Journal of Mathematics).
• An enriched count of the bitangents to a smooth plane quartic curve (Research in the Mathematical Sciences) [joint with Isabel Vogt].
• A refined Brill-Noether theory over Hurwitz spaces (Inventiones Mathematicae).
• Universal degeneracy classes for vector bundles on P1 bundles (Advances in Mathematics).
• Normal bundles of lines on hypersurfaces (Michigan Math Journal).
• Hyperbolicity of the partition Jensen polynomials (Research in Number Theory) [joint with Ian Wagner].
• Coefficients of McKay-Thompson series and distributions of the moonshine module (Proceedings of the AMS).
• Shifted distinct-part partition identities in arithmetic progressions (Annals of Combinatorics) [joint with Alwise et al].
• Proof of conjecture regarding the level of Rose's generalized sum-of-divisor functions (Research in Number Theory).
• Modular units from quotients of Rogers-Ramanujan type q-series (Proceedings of the AMS).
• Generalized Andrews-Gordon Identities (Research in Number Theory).
• Traces of singular values of Hauptmoduln (International Journal of Number Theory) [joint with Lea Beneish].
• Congruence properties of Taylor coefficients of modular forms (International Journal of Number Theory) [joint with Geoffrey Smith].
• Pseudo-unitary non-self-dual fusion categories of rank 4 (Journal of Algebra).

Upcoming conferences and talks:

• Department Colloquium, UC Berkeley, October 10, 2024
• AMS Fall Western Sectional, Invited Address and Special Session Co-organizer, UC Riverside, October 26 - 27, 2024
• Graduate Student Bootcamp, Mentor, Summer Research Institute in Algebraic Geometry, Colorado State University, July 8 - 12, 2025
• Geometry of Moduli, Summer Research Institute in Algebraic Geometry, Colorado State University, July 14 - 18, 2025

Presentation videos:

• Moduli Spaces, Bay Area Math Adventures (for high school students and the general public), Zoom, Fall 2024
Abstract: A moduli space is a space where each point corresponds to a shape of some kind. It helps us study all shapes of a given kind together. In this talk, we'll take a guided tour of the moduli space of circles and the moduli space of triangles. I hope the concept of moduli will give you a new perspective of these familiar shapes!
• Cohomology of moduli spaces of curves, Spec Q-bar(2 pi i), Fields Institute, Toronto, Summer 2024
Abstract: The moduli space M_g of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of Mg is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of Mg called the tautological ring. The definition of the tautological ring was later extended to the compactification Mg-bar and the moduli spaces with marked points M_g,n-bar. While the full cohomology ring of M_g,n-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups H^k(M_g,n-bar) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne.
• Chow rings of moduli spaces of pointed hyperelliptic curves, Spec Q-bar, Fields Institute, Toronto, Summer 2022
Abstract: In this talk, I describe the geometry of the moduli space H_{g,n} of n-pointed, genus g hyperelliptic curves. As n grows relative to g, work of Casnati and Schwarz shows that H_{g,n} goes from being rational (the simplest kind of variety) to general type (quite complicated). This suggests that we have hope of probing finer invariants of H_{g,n} when n is small relative to g. The Chow ring of H_{g,n} is one such invariant. I will describe an inductive procedure for stratifying H_{g,n} into nice pieces, which allows us to calculate its rational Chow ring when n is less than or equal to 2g + 6. This is joint work with Samir Canning.
• The rational Chow rings of M_7, M_8, and M_9, Derived Seminar, Winter 2021
Abstract: The rational Chow ring of the moduli space M_g of curves of genus g is known for g up to 6. In each of these cases, the Chow ring is tautological (generated by certain natural classes known as kappa classes). In recent joint work with Sam Canning, we prove that the rational Chow ring of M_g is tautological for g = 7, 8, 9, thereby determining the Chow rings by work of Faber. In this talk, I give an overview of our approach, with particular focus on the locus of tetragonal curves (special curves admitting a degree 4 map to P^1).
• Brill-Noether theory over the Hurwitz space, Western Algebraic Geometry Symposium (WAGS) Fall 2020
Abstract: Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I discuss joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.
• A refined Brill-Noether theory over Hurwitz spaces, Algebraic Geometry is Online in Zoom Everyone (AGONIZE), Spring 2020
Abstract: The celebrated Brill-Noether theorem says that the space of degree d maps of a general genus g curve to P^r is irreducible. However, for special curves, this need not be the case. Indeed, for general k-gonal curves (degree k covers of P^1), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I introduce a natural refinement of Brill-Noether loci for curves with a distinguished map C --> P^1, using the splitting type of push forwards of line bundles to P^1. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general k-gonal curves.

Teaching:

• Fall 2023, UC Berkeley, Math 143: Elementary Algebraic Geometry, Instructor
• Winter 2020, Stanford Math 51: Linear Algebra and Multivariable Calculus, Graduate Teaching Assistant
• Summer 2017, Emory University Number Theory Research Experience for Undergraduates, Instructor
• Spring 2016, Harvard Math 137: Algebraic Geometry, Undergraduate Course Assistant
• Spring 2015, Harvard Math 129: Algebraic Number Theory, Undergraduate Course Assistant