The main topics of my research are logarithmic geometry, deformation theory, moduli of curves, and Gromov-Witten theory. Below are brief descriptions of some of my papers. The list of my papers on the arXiv may be more current than the list below.
Preprints
The logarithmic Picard group and its tropicalization, with Samouil Molcho
constructs a canonical compactification of the Picard group of a nodal curve that is smooth, proper, possesses a group structure, but can only be represented in the category of logarithmic spaces. It is very naturally related to the tropical Picard group of a metric graph.
An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves, with Sebastian Casalaina-Martin
is an introduction to algebraic stacks. I particularly like the section on deformation theory at the end.
Stable maps to rational curves and the relative Jacobian and Logarithmic compactification of the Abel–Jacobi section, with Steffen Marcus
extend the notion of Cartier divisor to nodal curves so that the Abel–Jacobi map remains proper. The first paper uses orbifold methods and gets weaker results than the second one, which uses logarithmic and tropical geometry.
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.
Supported by NSF-MSPRF 0802951.
Published Papers
Moduli of stable maps in genus one and logarithmic geometry I and Moduli of stable maps in genus one and logarithmic geometry II, with Dhruv Ranganathan and Keli Santos-Parker
gives a modular means of contracting genus 1 subcurves of stable curves. We used this construction to give a smooth modular compactification of the moduli space of genus 1 curves in projective space (giving a modular interpretation to Vakil and Zinger's blowup construction). We also applied the same idea to stable quasimaps. The second paper uses the same ideas to address the realizability problem for tropical curves in genus 1. Notes from a talk.
Uniqueness of minimal morphisms of logarithmic schemes
shows that the space of morphisms between logarithmic schemes is quasifinite over the space of morphisms between the underlying schemes, under suitable hypotheses.
Project sponsored by the National Security Agency under Grant Number H98230-14-1-0107.
Invariance in logarithmic Gromov-Witten theory, with Dan Abramovich
proves that logarithmic Gromov-Witten invariants are invariant until logarithmic modifications.
Project sponsored by the National Security Agency under Grant Number H98230-14-1-0107.
Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, with Dan Abramovich and Steffen Marcus
proves the equivalence between several theories of Gromov-Witten invariants for smooth pairs (a smooth scheme with a smooth divisor) and double point degenerations (the union of two smooth schemes along a smooth divisor).
Supported by NSF-MSPRF 0802951.
Relative and orbifold Gromov-Witten invariants, with Dan Abramovich and Charles Cadman
shows that in genus zero, relative Gromov-Witten invariants are the same as orbifold Gromov-Witten invariants of a root stack.
Partly supported by NSF-MSPRF 0802951.
A hyperelliptic Hodge integral (to appear in Portugalia Mathematicae)
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.
Project partly supported by the National Security Agency under Grant Number H98230-14-1-0107.
Moduli of morphisms of logarithmic schemes
(to appear in Algebra & Number Theory)
constructs an algebraic space parameterizing the morphisms between two fixed logarithmic schemes.
Project sponsored by the National Security Agency under Grant Number H98230-14-1-0107.
Boundedness of the space of stable logarithmic maps, with Dan Abramovich, Qile Chen, and Steffen Marcus (to appear in J. Eur. Math. Soc.)
shows that the space of stable logarithmic maps to a fixed target is a disjoint union of quasi-compact components.
Project sponsored by the National Security Agency under Grant Number H98230-14-1-0107.
The deformation theory of sheaves of commutative rings II (Ann. Sc. Norm. Sup. Cl. Sci. (5) 14 (2015), no. 2)
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.
Supported by NSF-MSPRF 0802951.
Expanded degenerations and pairs
(Comm. Algebra 41 (2013) no. 6, 2346-2386)
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.
Supported by NSF-MSPRF 0802951.
The deformation theory of sheaves of commutative rings I
(J. Algebra, 352 (2012), 180-191)
is a quick introduction to the cotangent complex and its relationship to deformation theory. It avoids simplicial methods, relying heavily on Grothendieck topologies instead.
Supported by NSF-MSPRF 0802951.
Polynomial families of tautological classes on the space of curves with rational tails, with Renzo Cavalieri and Steffen Marcus
(J. Pure and Appl. Algebra 216 (2012), no. 4, 950-981)
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.)
Supported by NSF-MSPRF 0802951.
The genus zero Gromov-Witten invariants of [Sym2 P2] (Comm. Anal. Geom. 19 (2011), no. 5, 923-974)
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.
Talks
Brown University, November 1, 2019: Contracting genus 1 curves