 ## Shiyue Li 李詩樂

I am a mathematician at the Institute for Advanced Study, mentored by June Huh.

My research interest is 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 matroidal structures,

• applications of algebraic geometry to combinatorics.

In 2023, I got a PhD in mathematics at Brown University, where I was fortunate to be advised by Melody Chan. Prior, I had a wonderful time doing math and music at Harvey Mudd College, guided by the wisdom of Dagan Karp.

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

# Research

## Papers

1. Multimatroids and rational curves with cyclic action
(with Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang)
We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids introduced by Bouchet, which naturally arise in topological graph theory. The vantage point of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-A permutohedral varieties (Losev--Manin moduli spaces) and matroids, and the connection between type-B permutohedral varieties (Batyrev--Blume moduli spaces) and delta-matroids. pecifically, we equate a combinatorial nef cone of the moduli space with the space of ℝ-multimatroids, a slight generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, for the generating set of the Chow ring of the moduli space consisting of all psi-classes and their pullbacks along certain forgetful maps, we give a combinatorial formula for their intersection numbers by relating to the volumes of independence polytopal complexes of multimatroids.

2. Kapranov degrees
(with Joshua Brakensiek, Christopher Eur, Matt Larson)
The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of ψ-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding "Kapranov degrees," which include as special cases the work of Witten, Silversmith, Castravet-Tevelev, Postnikov, Cavalieri-Gillespie-Monin, and Gillespie-Griffins-Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.

3. Equivariant log-concavity of independence sequences of claw-free graphs
Séminaire Lotharingien de Combinatoire 89B.
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.

4. 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.

5. Wonderful compactifications and rational curves with cyclic action
Forum of Mathematics, Sigma 11. (2023).
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.

6. Equivariant log-concavity of graph matchings
Algebraic Combinatorics 6. (2023) .
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.

7. Intersecting psi-classes on tropical Hassett spaces
(with Marvin Anas Hahn)
Combinatorial Theory, 2(3). (2022).
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.

8. Permutohedral complexes and rational curves with cyclic action
manuscripta mathematica. (2022)
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.

9. Relative Bott--Samelson varieties
La Matematica 2. (2023)
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.

10. Topology of tropical moduli space of weighted stable curves in higher genus
(with Siddarth Kannan, Stefano Serpente, Claudia He Yun; slides)