• Three dimensional tropical correspondence formula. (CMP 2017, 26 pages) arXiv:1608.02306
    • Extends Mihkalkin's celebrated tropical correspondence formula for Gromov-Witten invariants to the three-dimensional case. This is the most interesting case, because curves of all genus contribute together — a vertex contributing n in Mikhankin's formula now contributes 2sin(nλ/2) to a generating function whose λ-exponent records Euler characteristic.

  • De Rham theory of exploded manifolds, (To appear in Geometry and Topology, 2017, 40 pages) arXiv:1003.1977
    • Most constructions for smooth manifolds work for exploded manifolds, almost without modification. De Rham cohomology is more delicate, because there are several different De Rham cohomology theories for exploded manifolds, all of which reduce to usual cohomology for smooth manifolds. In fact, these different cohomology theories are required for stating powerful tropical gluing formulas for Gromov-Witten invaraints.

  • Holomorphic curves in exploded manifolds, compactness. (Advances in Mathematics 2015, 72 pages) arXiv:0706.3917
    • The analogue of Gromov–compactness for holomorphic curves in exploded manifolds.

  • Exploded manifolds, (Advances in Mathematics, 229, 2012, 64 pages) arXiv:0910.4201
    • Read the introduction to this paper to learn why exploded manifolds are usefull to study Gromov-Witten invariants. As explained in this paper, most constructions familiar from differential geometry work for exploded manifolds.

  • Log geometry and exploded manifolds, (Abh. Math. Sem. Hamburg, 82, 2012, 39 pages) arXiv:1108.3713
    • If you know what a log scheme is, you should read this paper. It introduces exploded manifolds to log geometers, and explains the exploded-manifold perspective on log schemes — for example, it views nice log schemes are schemes over a paticular type of semi-ring. The included log-exploded dictionary suggests several directions for log geometry, and a tropical gluing formula for log Gromov-Witten invariants.

  • Exploded fibrations (Proceedings of Gokova geometry and topology conferece 2006, 2007, 39 pages) arXiv:0705.2408
    • This paper contains an earlier, more geometric construction of exploded manifolds and related spaces designed to study holomorphic curves in degenerations such as products of the degenerations that appear in symplectic field theory.

  • Holomorphic curves in Lagrangian torus fibrations, (Thesis, 2005, 117 pages) pdf
    • A cult hit. This thesis explains how holomorphic curves in Lagrangian torus fibrations can be recovered by studying graphs that appear in an adiabatic limit squeezing the torus fibers.

  • Holomorphic curves in exploded manifolds: Regularity (Submitted to Geometry and Topology, 2011, 48 pages) arXiv:0902.0087
    • This paper proves regularity for families of (perturbed) holomorphic curves in exploded manifolds. This inculdes a very strong version of the `gluing analysis' familiar to (pseudo)-holomorphic curve experts. The results of this paper are used — without no further analysis required — to prove the regularity of the moduli stack of holomorphic curves in exploded manifolds in the paper `Holomorphic curves in exploded manifolds: Kuranishi structure' below.

  • Gromov-Witten invariants of exploded manifolds, (102 pages, 2011) arXiv:1102.0158v1
    • I have rewritten parts of this paper in order to allow the construction of integer valued invariants, and to make the construction of Gromov-Witten invariants more natural and easier to understand.


  • Holomorphic curves in exploded manifolds: Kuranishi structure, (86 pages, 2013) arXiv:1301.4748
    • This paper constructs a smooth Kuranishi structure on the moduli stack of holomorhpic curves in an exploded manifold. The Kuranishi structure constructed is more concrete than the Kuranishi structures introduced by Fukaya and Ono because it is naturally embedded in a moduli stack of (not neccesarily holomorphic) curves.

  • Universal tropical structures for curves in exploded manifolds, (32 pages, 2013) arXiv:1301.4745
    • An important step in the analysis of the moduli stack of holomorphic curves in an exploded manifold is to construct a family of curves with `universal tropical structure', as done in this paper.

  • Integral counts of pseudo-holomorphic curves, (75 pages, 2013) arXiv:1309.0585
    • This paper constructs an integer-valued invariant counting holomorphic curves in any compact symplectic manifold. The definition was originally suggested by Fukaya and Ono, however a key step requires the Kuranishi structure constructed in `Holomorphic curves in exploded manifolds: Kuranishi structure'. The relationship between these invariants and Taubes' Gromov invariants of symplectic 4-manifolds is explained in the introduction.

  • On the value of thinking tropically to understand Ionel's GW invariants realtive normal crossing divisors, (17 pages, 2014) arXiv:1407.3020
    • This nontechnical paper works through a simple example with lots of pictures to explain the relationship between Ionel's GW invariants relative normal-crossing divisors and Gromov-Witten invariants of exploded manifolds.

  • Tropical enumeration of curves in blowups of the projective plane. PDF, Mathematica notebook GWblowupsofplane.nb, data gw, fancy talk slides
    • This nontechnical paper explains how a tropical gluing formula allows the computation of Gromov-Witten invariants of blowups of the projective plane. The computation is via Gromov-Witten invariants relative a certain normal crossing divisor. These relative Gromov-Witten invariants may be calculated recursively, then used to specify the absolute Gromov-Witten invariants. The resulting recursive method for calculating Gromov-Witten invariants is different from the known methods by Goettsche and Pandharipande in the zero genus case, and by Caporaso and Harris in the case of invariants of the projective plane (with no blowups.)
    • To use the program GWblowupsofplane.nb, download it and open it with mathematica. Also download gw, which is the result of running calculations from GWblowupsofplane.nb for a day. With gw, you can get Gromov-Witten invariants of any blowup of the projective plane up to degree 12 instantly. For higher degree invariants, you will need to follow instructions in GWblowupsofplane.nb (and wait a long time.)

  • Gluing formula for Gromov-Witten invariants in a triple product, (15 pages, 2015.) arXiv:1511.00779
    • This non-technical paper explains the tropical gluing formula for Gromov-Witten invariants in a triple product, and gives some short examples. This simple example is chosen so that the gluing formula can be explained with minimal reference to exploded manifolds.

  • Holomorphic curves in exploded manifolds: virtual fundamental class, (54 pages, 2015.) arXiv:1512.05823
    • This paper defines Gromov-Witten invariants of exploded manifolds using embedded Kuranishi structures constructed in arXiv:1301.4748. The technical heart of this paper is a construction of a virtual fundamental class of any Kuranishi category — a simplified version of an ebedded Kuranishi structure. We also show how to integrate differential forms over this virtual fundamental class to obtaine numerical invariants, and push forward differential forms over evaluation maps. We show that such invariants do not depend on any choices in the construction, and are compatible with pullbacks, products, and tropical completion of Kuranishi categories. In the case of a compact symplectic manifold, this paper gives an alternate construction of Gromov-Witten invariants, including gravitational descendants.

  • Notes on exploded manifolds and a tropical gluing formula for Gromov-Witten invariants arXiv:1605.00577
    • Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants, including a degeneration formula for Gromov-Witten invariants in normal crossing degenerations. I gave the original lecture series in April 2016 at the Simons Center for Geometry and Physics at Stonybrook. Video of the lectures is available on the SCGP website at this URL.

  • Tropical gluing formulae for Gromov-Witten invariants, (41 pages, 2017.) arXiv:1703.05433
    • This paper proves tropical gluing formulae for Gromov-Witten invariants, useful for calculating Gromov-Witten invariants of a symplectic manifold given a normal-crossing degeneration. The same formuale apply to Gromov-Witten invariants relative normal-crossing divisors, and provide a rich structure for such invariants.