Shiyue Li 詩樂

I’m a PhD candidate in Mathematics at Brown University. I’m fortunate to be advised by Melody Chan.

My research interest is in combinatorial algebraic geometry. More specifically, I think about K-theory, Hodge structures and intersection theory of moduli spaces of curves, geometric models and invariants of matroids, and applications of algebraic geometry to combinatorics.

From fall 2023, I will be a member at the Institute for Advanced Study, mentored by June Huh.

Prior, I was at Yale University and Harvey Mudd College, doing math and (an excessive amount of) music. My undergrad thesis was advised by Dagan Karp.

Here are my Curriculum Vitae, Google Scholar, arXiv and Email.

Research

Papers

  1. Equivariant log-concavity of independence sequences of claw-free graphs
    We show that the graded vector space spanned by independent vertex sets of any claw-free graph is strongly equivariantly log-concave, viewed as a graded permutation representation of the graph automorphism group. Our proof reduces the problem to the equivariant hard Lefschetz theorem on the cohomology of a product of projective lines. Both the result and the proof generalize our previous result on graph matchings. This also gives a strengthening and a new proof of results of Hamidoune, and Chudnovsky--Seymour.

  2. K-rings of wonderful varieties and matroids
    (with Matt Larson, Sam Payne, Nick Proudfoot)
    We study the K-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We give a new invariant of arbitrary matorids. It computes the Euler characteristics of arbitrary line bundles on wonderful varieties and on Deligne--Mumford--Knudsen moduli spaces of stable rational curves with marked points.

  3. Wonderful compactifications and rational curves with cyclic action
    (with Emily Clader, Chiara Damiolini, Rohini Ramadas)
    To appear in Forum of Mathematics, Sigma.
    We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

  4. Equivariant log-concavity of graph matchings
    To appear in Algebraic Combinatorics.
    We show that the graded vector space spanned by matchings in any graph is strongly equivariantly log-concave, viewed as a graded permutation representation of the graph automorphism group. Our proof constructs equivariant injections by reducing to the hard Lefschetz theorem.

  5. Intersecting psi-classes on tropical Hassett spaces
    (with Marvin Anas Hahn)
    Combinatorial Theory, 2(3), https://doi.org/10.5070/C62359165.
    We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig. Our computation reveals that the weight of a maximal cone in an intersection has a combinatorial intepretation in terms of the underlying tropical curve and it is always nonnegative. In particular, our result specialises to that, in top dimension, the tropical intersection product coincides with its classical counterpart.

  6. Permutohedral complexes and rational curves with cyclic action
    (with Emily Clader, Chiara Damiolini, Daoji Huang, Rohini Ramadas)
    manuscripta mathematica. DOI: 10.1007/s00229-022-01419-6
    We construct a moduli space of rational curves with finite-order automorphism and weighted orbits, and studied a three-way bijection between the boundary strata of the moduli space, the faces of a polytopal complex, and certain cosets of the complex reflection group.

  7. Relative Bott-Samelson varieties
    We generalize Brion's Bott-Samelson resolution of singularities of Richardson varieties to a relative setting. We then applied it to give a resolution of singularities for the Brill-Noether variety with imposed ramification on twice-marked elliptic curves, studied by Chan--Osserman--Pflueger.

  8. Topology of tropical moduli space of weighted stable curves in higher genus
    (with Siddarth Kannan, Stefano Serpente, Claudia He Yun; slides)
    To appear in Advances in Geometry.
    We prove the simple connectivity of the moduli spaces of weighted stable tropical curves of volume one, for all higher genus and all rational weights. We also compute their Euler characteristics.

  9. Chow ring of heavy/light Hassett spaces via tropical geometry
    (with Siddarth Kannan and Dagan Karp)
    Journal of Combinatorial Theory, Series A. 178C (2021) 105348
    We compute the Chow ring of heavy/light Hassett spaces of genus zero via tropical compactification and toric intersection theory.

Invited talks and travels

Teaching

Canada/USA Mathcamp