The moduli M_{g,n} of genus-g n-pointed algebraic curves famously has a "tropicalization" M_{g,n}^{trop} that parametrizes graphs. In this paper, we explore how the relationship between M_{g,n} and M_{g,n}^{trop} lifts to non-algebraic spaces that live over M_{g,n}. For example, we find that Culler-Vogtmann's Outer space, introduced in the 1980s to study outer automorphisms of free groups, is essentially the tropicalization of a moduli space of "complex handlebodies". We end up with a nice web of relationships between "complex" and "tropical" spaces that fits together many familiar objects from geometric group theory and surface topology, including Harvey's curve complex, mapping class groups of surfaces and handlebodies, the moduli space of Schottky groups, and augmented Teichmüller space.

We introduce an intersection number on the moduli space of stable curves M0,n-bar (or more generally on Mg,n-bar) associated to a finite simple graph G. We prove that these numbers are equal to evaluations of the chromatic polynomial of G at negative integers. Actually we give two proofs: One is based on the deletion-contraction formula and involves classical intersection theory of moduli spaces of curves, while the other relates these intersection numbers to the theory of hyperplane arrangements and maximum likelihood degrees.

This short paper gives a simple closed formula for a class of cross-ratio degrees related to the theory of positive geometries and to the computation of string amplitudes. The main result is entirely concrete: these degrees are indexed by triangulations T of an n-gon, and the associated degree is 2^{I(T)}, where I(T) is the number of internal triangles in T.

We study the T-graph of the Hilbert scheme Hilbn(C2) of finite-length subschemes of the plane --- a graph that encodes the connectivity of torus orbits.  A complete combinatorial description of this graph is a (probably-impossible) open problem -- we give a natural subgraph that we can fully understand via tropical ideals.

This paper accomplishes two things. First, we introduce a tropical tool for studying cohomological field theories, data structures arising in string theory that encode Gromov-Witten theory as well as other related theories, like Witten's r-spin theory. Namely, we show that the balancing conditions on genus-zero tropical CohFT cycles are equivalent to the usual WDVV relations. Second, we use this tool to simplify the combinatorics of genus-zero r-spin invariants significantly, ultimately producing (Equation 1) a very simple closed formula for all genus-zero r-spin invariants! We also prove other combinatorial properties of r-spin invariants, e.g. that they respect the natural (dominance) order on the inputs.

Critical-orbit-relation varieties are moduli spaces of rational self-maps of P1 with specified behavior of the forward orbits of critical points. These spaces are non-compact --- we develop techniques for studying their geometry via the admissible covers compactification, and give two applications. Ramadas later used these tools to prove an implication between two long-standing open problems -- irreducibility of Gleason polynomials over Q, and irreducibility of Pern(0) over C.

The homology groups of M0,n-bar are well-studied. H2k(M0,n-bar) carries a natural Sn-action, and surprisingly, it is open whether the action has an invariant basis. This is known to hold for k=1 (Gibney-Farkas), and for the entire ring H*(M0,n-bar) (Castravet-Tevelev). We show that it holds for k=2. Later, Kiem-Choi-Lee showed it holds for k=3.

Cross-ratio degrees are a natural class of intersection numbers on the moduli space M0,n-bar of stable rational curves, indexed by certain bipartite graphs. While no combinatorial expression is known for them, I show (via geometry!) that they are bounded above by counts of perfect matchings of the bipartite graphs in question. This is one of my favorite projects.

We study the Chow rings of moduli spaces of sheaves on a K3 surface. We define and study the "tautological subring", and conjecture several nice properties that would follow from expected identities in the Chow ring --- based on the modified diagonals of O'Grady and Voisin. We prove our conjecture for Hilbert schemes of points on a K3 surface.

I study the "tropical stratification" of Hilbn(P1). To a subscheme of P1, we can associate a combinatorial object called its tropical ideal, and divide up Hilbn(P1) by which subschemes have the same tropical ideal. I show that the equations defining the stratification are precisely the Schur polynomials in n variables. I discuss special strata associated to binary necklaces.

This is my PhD thesis work. I give an algorithm for computing genus-zero Gromov-Witten invariants of Symd(Pr). I apply the algorithm to prove a mirror theorem for Symd(Pr). The proof is based on Coates-Corti-Iritani-Tseng's mirror theorem for toric stacks, but several new tools were needed to deal with the fact that Symd(Pr) is not toric (or even abelian).