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). Also available at arXiv:2011.13903.

2. Artin-Schreier root stacks. Journal of Algebra, vol. 586 (2021), 1014 - 1052. Also 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). Also available at doi:10.1142/S0218216518500566 and arXiv:1405.7683.

Preprints

4. Artin-Schreier-Witt theory for stacky curves (2023). Also available at arXiv:2310.09161.

3. Categorifying zeta functions of hyperelliptic curves, with J. Aycock (2023). Also available at arXiv:2304.13111.

2. Categorifying quadratic zeta functions, with J. Aycock (2022). Also available at arXiv:2205.06298.

1. A¹-local degree via stacks, with L. Taylor (2020). Also available at arXiv:1911.05955.

Other writing

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

Projects

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

5. Zeta functions and decomposition spaces, with Jon Aycock (in progress). So far has produced the article “A primer on zeta functions and decomposition spaces” (see Publications) and the preprints “Categorifying quadratic zeta functions” and “Categorifying zeta functions of hyperelliptic curves” (see Preprints).

4. A¹-local degree via stacks, with Libby Taylor. Produced the preprint “A¹-local degree via stacks” (see Preprints).

3. Wild Ramification and Stacky Curves. PhD dissertation at University of Virginia. Adviser: Andrew Obus. Available at doi:10.18130/v3-y9s6-kt47. So far has produced the article “Artin-Schreier root stacks” (see Publications) and the preprint “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. Culminated in the paper “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.