Equivariant Gromov-Witten theory of stacky P1s.
My thesis, which studies the equivariant Gromov-Witten theory of one dimensional toric stacks. This is the starting point of a larger project to study the GW invariants of all stacky curves. The stack can be ineffective - that is, the generic point can have an isotropy group. For example, consider M1,1: every genus 1 curve has an involution. The main result is an operator formula for the GW invariants in terms of a Fock space for wreath products. This formula leads has nice consequences: a proof that the GW theory of ineffective orbifolds decomposes, and a proof that the GW invariants satisfy a form of the 2-Toda hierarchy.
Abelian Hurwitz-Hodge integrals,
Rahul Pandharipande and
Hsian-Hua Tseng .
The ELSV formula expresses certain integrals over the moduli space of curves in terms of Hurwitz numbers, and has been a vital tool in Gromov-Witten theory. This paper gives an analagous formula for moduli spaces of orbifold curves, and is the key tool in my thesis.
Tropical Hurwitz Numbers,
We show that tropical double Hurwitz numbers agree with classical double Hurwitz numbers. In the process, we discover a new method for calculating double Hurwitz numbers in terms of graphs, and apply this to get easy proofs of polynomiality and wall crossing in genus zero.
Chamber Structure for double Hurwitz numbers, joint with
A continuation and application of the previous work: we use the organization of the cut and join equation into graphs to study polynomiality and wall crossing for higher genus double Hurwitz numbers. To prove wall crossing, we use the combinatorial Gauss-Manin connection machinery of Varchenko, and prove a combinatorial formula for the Gauss-Manin connection for cographic arrangements. The method seems to point to a connection with the stability conditions of Seshadri and Oda for compactifying Jacobians.
Double Hurwitz numbers via the infinite wedge.
Following Okounkov, the expression for double Hurwitz numbers in terms of representation theory of the symmetric group is conveniently packaged on the infinite wedge, which leads to closed form expressions for certain generating functions. This in turn, yield strong results about the polynomiality of double Hurwitz Numbers; in particular, a proof of the Strong Piecewise Polynomiality Conjecture of Goulden, Jackson, and Vakil. The paper begins with a introduction to the needed results about the infinite wedge. While the polynomiality results obtained are stronger than in the previous papers, the methods are completely algebraic, and so the tantalizing hints to geometry are gone.