Research

 
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Research Interests

Click here to read my research statement.

My research lies in three main areas: arithmetic geometry, number theory and homotopy theory. A unifying theme is the theory of algebraic stacks. For me, these are essential tools for understanding some of the following topics:

  • Arithmetic geometry: covers of curves, wild ramification, étale fundamental groups, formal orbifolds, local-global principles;

  • Algebraic geometry: moduli problems, stacky curves;

  • Number theory: modular curves, modular forms, zeta and L-functions, generalized Fermat equations;

  • Homotopy theory: A¹ (or “enriched”) enumerative geometry, homotopy theory of stacks, objective zeta functions, decomposition spaces/2-Segal spaces, incidence algebras, homotopy linear algebra.

One of my main projects is to bring techniques from finite characteristic arithmetic geometry (e.g. Artin-Schreier-Witt theory) into the world of stacks. This has opened a number of new research directions, including applications to modular forms and Galois representations.

Publications

3. A primer on zeta functions and decomposition spaces. Moduli, Motives and Bundles - New Trends in Algebraic Geometry, London Mathematical Society Lecture Notes Series (2026), 233 - 264. Available at ISBN:9781009497190 and arXiv:2011.13903.

2. Artin-Schreier root stacks. Journal of Algebra, vol. 586 (2021), 1014 - 1052. Available at doi.org/10.1016/j.jalgebra.2021.07.023 and arXiv:1910.03146.

1. Crossing number bounds in knot mosaics, with H. Howards. Journal of Knot Theory and its Ramifications, vol. 27, no. 10 (2018). Available at doi:10.1142/S0218216518500566 and arXiv:1405.7683.

Preprints

7. Wild stacky curves and rings of mod p modular forms, with D. Zureick-Brown (in preparation).

6. The integral Hasse principle for stacky curves associated to a family of generalized Fermat equations, with J. Duque-Rosero, C. Keyes, M. Roy, S. Sankar and Y. Wang (in preparation).

5. Arithmetic functions and geometry (2025). Available at arXiv:2504.13328.

4. Categorifying zeta functions for quadratic covers, with J. Aycock (2025). Available at arXiv:2304.13111.

3. Artin-Schreier-Witt theory for stacky curves (2023). Available at arXiv:2310.09161.

2. Categorifying zeta functions of hyperelliptic curves, with J. Aycock (2023). Available at arXiv:2304.13111v1.

1. Categorifying quadratic zeta functions, with J. Aycock (2022). Available at arXiv:2205.06298.

Other writing

2. “How to Have Lunch in the Time of COVID-19”, with K. DeVleming. Notices of the AMS (January 2021).

1. A¹-local degree via stacks, with L. Taylor (2020). Withdrawn from arXiv after errors were identified. A corrected version may be drafted at a later date. If you are interested in the details in this project, email me or Libby!

Projects

10. Stacky curves and generalized Fermat equations, with Santiago Arango-Piñeros and David Zureick-Brown (in progress).

9. Supersingular mass formulas for abelian varieties, with Eran Assaf (in progress).

8. Supersingular mass formulas for modular curves, with Santiago Arango-Piñeros and Sun Woo Park (in progress).

7. Local-global principle for stacky curves, with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang (in progress). In this project, we use étale descent theory on a stacky curve to characterize local and global solutions to spherical generalized Fermat equations, which leads to asymptotic results on the number of such stacky curves satisfying the local-global principle for integral points. Produced:

  • “The integral Hasse principle for stacky curves associated to a family of generalized Fermat equations” (see Preprints).

6. Rings of mod p modular forms, with David Zureick-Brown (in progress). In this project, we use the stacky structure of modular curves in characteristic p (see Project 3) to give a presentation for rings of mod p modular forms. We also characterize ethereal modular forms geometrically. Produced:

  • “Wild stacky curves and rings of mod p modular forms” (see Preprints).

5. Objective zeta and L-functions, with Jon Aycock (in progress). In this project, we aim to unify various types of zeta and L-functions using Gálvez-Carrillo, Kock and Tonks’ theory of objective incidence algebras. Produced:

  • “Arithmetic functions and geometry” (see Preprints);

  • “Categorifying zeta functions for quadratic covers” (see Preprints);

  • “Categorifying zeta functions of hyperelliptic curves” (see Preprints);

  • “Categorifying quadratic zeta functions” (see Preprints);

  • “A primer on zeta functions and decomposition spaces” (see Publications).

4. A¹-local degree via stacks, with Libby Taylor. In this project, we investigated Levine, Kass and Wickelgren’s A¹-enumerative geometry program in the context of algebraic stacks. Produced:

  • “A¹-local degree via stacks” (see Other Writing).

3. Wild Ramification and Stacky Curves. PhD dissertation at University of Virginia. Adviser: Andrew Obus. Available at doi:10.18130/v3-y9s6-kt47. In this thesis, I characterized the stacky structure of wild stacky curves in characteristic p > 0 and obtained results about canonical rings of such curves. Culminated with:

  • “Artin-Schreier-Witt theory for stacky curves” (see Preprints);

  • “Artin-Schreier root stacks” (see Publications).

2. Class Field Theory and the Study of Symmetric n-Fermat Primes. Master’s thesis at Wake Forest University. Adviser: Frank Moore. In this thesis, I surveyed class field theory and computed data indicating asymptotic trends in pairs of primes represented symmetrically by diagonal quadratic forms.

1. Saturation in Knot Mosaics. Senior thesis at Wake Forest University. Adviser: Hugh Howards. In this thesis, we bounded the number of crossings of a knot admitting a mosaic representation in terms of the size of the mosaic. Produced:

  • “Crossing number bounds in knot mosaics” (see Publications).

Course Notes

I keep notes on many courses and seminars I have taught or been a part of, which you can find here.