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.
Preprints and papers under review
- Twice-Marked Banana Graphs & Brill-Noether Generality (with N. Solomon). Preprint (2022).
arXiv - 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 - Versality of Brill-Noether flags and degeneracy loci of twice-marked curves. Preprint (2021).
arXiv
Publications
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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 p. 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
- 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.