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 persepctives via non-Archimedean analytic geometry. My publications are listed below.

Papers

  • Relative Richardson Varieties (with M. Chan). Preprint. arXiv
  • Combinatorial relations on skew Schur and skew stable Grothendieck polynomials (with M. Chan). To appear in Algebraic Combinatorics. arXiv
  • 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
  • Euler characteristics of Brill-Noether varieties (with M. Chan). Transactions of the AMS 374:3 (2021) 1513-1533. arXiv
  • Weierstrass semigroups on Castelnuovo curves. To appear in Journal of Algebra. arXiv
  • On non-primitive Weierstrass points. Algebra and Number Theory 12 (2018), no. 8, 1923-1947. arXiv
  • Brill-Noether varieties of k-gonal curves. Advances in Mathematics 312 (2017) 46-63. arXiv. A video of me speaking about this result is here.
  • Special divisors on marked chains of cycles. Journal of Combinatorial Theory, Series A 150 (2017) 182-207. arXiv
  • 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
  • Bitangents of tropical plane quartic curves (with M. Baker, Y. Len, R. Morrison, and Q. Ren). Mathematische Zeitschrift 282:3 (2016) 1017-1031. arXiv
  • On linear series with negative Brill-Noether number. Unpublished manuscript. arXiv
  • Graph reductions, binary rank, and pivots in gene assembly. Discrete Applied Mathematics 159:17 (2011) 2117-2134. arXiv

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 of 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.

Source code

  • Enumeration of semigroups of maximum effective weight: source, output