Obstruction theories and virtual fundamental classes
gives a new definition of an obstruction theory closely related to an earlier definition of Li and Tian and proves it is equivalent, in certain circumstances, to another definition of Behrend and Fantechi.
The deformation theory of sheaves of commutative rings II
shows that some of the first properties of the cotangent complex can be proved using Grothendieck topologies in place of simplicial rings. It also contains a comparison between the obstruction classes obtained by these methods and those originally defined by Illusie.
A hyperelliptic Hodge integral
is another paper from my thesis. It contains the evaluation of a particular family of integrals on a compactification of the space of hyperelliptic curves that turns out to be important for relating orbifold Gromov-Witten invariants to the enumerative geometry of hyperelliptic curves.
Published Papers
Expanded degenerations and pairs
(to appear in Comm. Algebra)
is about an interesting stack and a number of moduli problems that it solves: among them are moduli of semistable genus zero curves, moduli of expanded degenerations and pairs (as introduced by J. Li), moduli of aligned logarithmic structures, and moduli of sequences of homomorphisms of line bundles.
The deformation theory of sheaves of commutative rings I
(J. Algebra)
is a quick introduction to the cotangent complex and its relationship to deformation theory. It avoids simplicial methods, but relies heavily on Grothendieck topologies.
Polynomial families of tautological classes on the space of curves with rational tails, with Renzo Cavalieri and Steffen Marcus
(J. Pure and Appl. Algebra)
shows the equivalence of two tautological classes on the space of rational tails curves: the first class is the space of relative stable maps to a "rubber" rational target and the second is the vanishing locus of a section of the universal Jacobian. The comparison makes it possible to evaluate some of these classes explicitly. (One of the classes in this comparison above has been evaluated more generally by Richard Hain.)
The genus zero Gromov-Witten invariants of [Sym2P2] (to appear in Comm. Anal. Geom.)
is my thesis. It contains a calculation of the orbifold Gromov-Witten invariants of the stack symmetric square of the projective plane and relates some of them to the enumerative geometry of hyperelliptic curves in the plane. It was also one of the first non-toric examples of the crepant resolution conjecture, at least at one time.
Some videos from an undergraduate talk I gave about moduli spaces.
A proof of the snake and salamander lemmas in an arbitrary abelian category, without recourse to elements via Mitchell's embedding theorem. I suppose this might be called a non-elementary proof, though it is quite easy.
