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 principle

  • Algebraic geometry: moduli problems, stacky curves, Bertini theorems

  • 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 (to appear). Available at 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

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 Withdrawn from arXiv after errors were identified.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).

6. Rings of mod p modular forms, with David Zureick-Brown (in progress).

5. Objective zeta and L-functions, with Jon Aycock (in progress). Produced:

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

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

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

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

4. A¹-local degree via stacks, with Libby Taylor. 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. Culminated with:

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

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

2. Class Field Theory and the Study of Symmetric n-Fermat Primes. Master’s thesis at Wake Forest University. Adviser: Frank Moore.

1. Saturation in Knot Mosaics. Senior thesis at Wake Forest University. Adviser: Hugh Howards. 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.