Research
My research is in algebraic geometry and combinatorics, especially algebraic curves and Young tableaux. I am particularly interested in Brill–Noether theory, which studies the variety of ways that a chosen curve is realized in projective space. I also study analogous questions in the setting of tropical geometry, and the relation between these perspectives via non-Archimedean analytic geometry. My publications are listed below.
I am also interested in proof formalization in Lean 4. You can find some of my ongoing formalization projects below.
Lean 4 formalizations
Note: Some of these formalizations have links to documentation. This documentation is auto-generated and includes all dependencies within Mathlib. When reading the documentation, look for the relevant section in the sidebar and focus on those files.
- Brill–Noether theory of graphs. This repository is mainted by generative AI, and therefore should be regarded as experimental. It contains formalizations of several results from the literature about special divisors on finite graphs, and some novel results as well. Some results from this repository are registered on Palomar.
github -
Chip-firing in Lean (with Dhyey Mavani). This project formalizes aspects of chip-firing games on finite multigraphs, including the Riemann–Roch theorem of Baker and Norine.
github docs arXiv Palomar - Demazure products. This project formalizes the extended Demazure product on integer permutations (not necessarily of finite length). It includes a formalized bijection between Hecke factorizations of integer permutations and set-valued tableaux.
github docs
Preprints and papers under review
- An extended Demazure product on integer permutations via min-plus matrix multiplication. Preprint (2022).
arXiv - (Hurwitz–)Brill–Noether general marked graphs via the Demazure product. Preprint (2022).
arXiv - chipfiring: A Python Package for Efficient Mathematical Analysis of Chip-Firing Games on Multigraphs (with Tairan Ji and Dhyey Mavani). Preprint (2025).
arXiv
Publications
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Transmission permutations and Demazure products in Hurwitz–Brill–Noether theory. To appear in International Mathematics Research Notices.
arXiv DOI -
Twice-Marked Banana Graphs & Brill–Noether Generality (with N. Solomon). Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1415-1457.
arXiv DOI -
Versality of Brill–Noether flags and degeneracy loci of twice-marked curves. Algebraic Geometry 12:4 (2025) pp. 497-518.
arXiv DOI -
Weierstrass semigroups from cyclic covers of hyperelliptic curves (with E. Cotterill and N. Zhang). Bulletin of the Brazilian Mathematical Society, New Series 54, 37 (2023).
arXiv DOI - Linear series with ρ<0 via thrifty lego-building. Journal für die reine und angewandte Mathematik (Crelle), no. 797 (2023), pp. 193-228.
arXiv DOI - Relative Richardson Varieties (with M. Chan). Mathematical Proceedings of the Cambridge Philosophical Society, 175:1 (2023), 161–186.
arXiv DOI - Combinatorial relations on skew Schur and skew stable Grothendieck polynomials (with M. Chan). Algebraic Combinatorics Volume 4 (2021) no. 1 pp. 175-188.
arXiv DOI pdf - Euler characteristics of Brill–Noether varieties (with M. Chan). Transactions of the AMS 374:3 (2021) 1513-1533.
arXiv DOI pdf - Weierstrass semigroups on Castelnuovo curves. Journal of Algebra 582 (2021) 117-135.
arXiv DOI pdf - The Gieseker–Petri theorem and imposed ramification (with M. Chan and B. Osserman). Bulletin of the London Mathematical Society 51:6 (2019) 945-960.
arXiv DOI pdf - On non-primitive Weierstrass points. Algebra and Number Theory 12 (2018), no. 8, 1923-1947.
arXiv DOI pdf - Genera of Brill–Noether curves and staircase paths in Young tableaux (with M. Chan, A. López Martín, and M. Teixidor i Bigas). Trans. Amer. Math. Soc. 370 (2018) 3405-3439.
arXiv DOI pdf - Brill–Noether varieties of k-gonal curves. Advances in Mathematics 312 (2017) 46-63.
arXiv DOI pdf - Special divisors on marked chains of cycles. Journal of Combinatorial Theory, Series A 150 (2017) 182-207.
arXiv DOI pdf - Bitangents of tropical plane quartic curves (with M. Baker, Y. Len, R. Morrison, and Q. Ren). Mathematische Zeitschrift 282:3 (2016) 1017-1031.
arXiv DOI pdf - Graph reductions, binary rank, and pivots in gene assembly. Discrete Applied Mathematics 159:17 (2011) 2117-2134.
arXiv DOI pdf
Exposition
- Tropical Curves. pdf Expositio
- This is my minor thesis from graduate school. It is an introduction to tropical curves, with an emphasis on how they are analogous to algebraic curves, both formally and informally.
- The minor thesis is part of the Harvard graduate program in which students must rapidly learn (in three weeks) a subject outside their research and write an expository paper about it. This was a very useful exercise for me, and in fact tropical curves later became an important part of my research!
- Because this paper was created on a short deadline, it is not particularly polished, and (for example) has hand-drawn diagrams. Nonetheless, I am happy to make it available online for anyone who might benefit from it.