Research

I am currently on sabbatical, therefore currently immersed in research.

Publications can be found below.

When not on sabbatical I organize a weekly geometry seminar, generally in tandem with Dr. Kenneth Hughes' analytic number theory seminar.

I am currently working, in a more or less coherent and organized fashion, on the following projects:

• Understanding generic stable maps from smooth 4-manifolds to the 2-sphere, 1-parameter families of such maps, and the extent to which such maps can shed new light on smooth 4-manifold invariants. This is joint work with Rob Kirby and is a natural continuation of our work on constructions of Lefschetz-type fibrations. All of our ideas are very much inspired by and built upon the work of Auroux-Donaldson-Katzarkov, Perutz, Lekili, Baykur and Williams.
• Liberating toric and tropical ideas from their algebro-geometric shackles and hoping that they can tell us something interesting about smooth 4-manifolds. This is work with Margaret Symington and naturally flows out of our earlier collaboration on Lagrangian fibrations of near-symplectic 4-manifolds. In particular we would like to understand how to use singular integral affine structures on 2-complexes to describe smooth, symplectic or near-symplectic 4-manifolds. The long-term optimistic fantasy is that this might help with enumerating pseudo-holomorphic curves and thus evaluating smooth 4-manifold invariants, bringing together ideas of Taubes and Mikhalkin. In the short term we are still trying to understand, from a smooth and symplectic topology point of view, precisely how Mikhalkin's higher-dimensional pair of pants (in dimension 4 this is the complex projective plane minus 4 generic complex projective lines) fibers over its corresponding tropical version (which, in the dimension we care about, is topologically the cone on the 1-skeleton of a tetrahedron). If anyone can explain this to us in simple smooth topology terms, we would love to hear about it. On the enumerative geometry side, we would almost certainly need to use (and understand) the work of Brett Parker.
• Further generalizing the rational blowdown techniques that have been so useful in recent constructions of small exotic 4-manifolds, and making sure that they work in the symplectic setting. This is joint work with Andras Stipsicz, and we have already made significant progress. This is closely related to smoothings and resolutions of algebraic singularities, classification of tight contact structures on 3-manifolds, and many of the toric (tropical?) ideas that come out of Symington's work.

The speculative zone

These are some of my much less coherent research questions, thoughts and plans:

• Understand the oriented graph whose vertices are open book decompositions and whose edges are plumbings (one version allows left- and right-handed plumbings, one version only allows right-handed plumbings).
• Is there a direct proof of Harer's conjecture that doesn't rely on contact topology?
• Given an open book decomposition on a connected sum of n copies of (S^1 X S^2) which supports a contact structure with c_1 = 0, can one find n 1-handles in the page across each of which the monodromy is trivial? I.e. can one find n properly embedded arcs which do not, collectively, disconnect the page, such that the monodromy is trivial on each of these arcs? If so, then this OBD comes from an OBD of S^3 by n standard 1-surgeries. I suppose I have no evidence that this is true other than the absence of counterexamples.
• Find an example of an OBD on S^3 which is obtained from the trivial D^2 OBD by plumbing on n_+ right-handed Hopf bands and m_+ left-handed Hopf bands, then deplumbing n_- right-handed Hopf bands and m_- left-handed Hopf bands, such that n_- > n_+. Actually, I know an example of such an OBD, but the problem is to find the sequence of plumbings and deplumbings explicitly, rather than just believing Giroux-Goodman.
• In my thesis I defined a framing-valued thickness of transverse knots and links in contact 3-manifolds. I have no idea how to compute it in general, except to get some obvious bounds.

• ``Toric structures on near-symplectic 4-manifolds'', with M. Symington, math.SG/0609753. To appear in J. Eur. Math. Soc.
• ``Constructing Lefschetz-type fibrations on four-manifolds'', with R. Kirby, Geom. Topol. 11 (2007) 2075-2115. Available electronically here.
• David T. Gay and András I. Stipsicz ``Symplectic Rational Blow-down Along Seifert Fibered 3-Manifolds'', with A. Stipsicz, Int Math Res Notices , 2007 Volume 2007: article ID rnm084, 20 pages. Available electronically here.
• ``Four-dimensional symplectic cobordisms containing three-handles'', Geom. Topol. 10 (2006) 1749-1759. Available electronically here.
• ``Constructing symplectic forms on 4-manifolds which vanish on circles'', with R. Kirby, Geom. Topol. 8 (2004) 743-777. Available electronically here.
• ``Open books and configurations of symplectic surfaces'', Algebr. Geom. Topol. 3 (2003), 569-586. Available electronically here.
• ``Explicit concave fillings of contact three-manifolds'', Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 3, 431-441. See also http://arXiv.org, arXiv:math.GT/0104059.
• ``Symplectic 2-handles and transverse links'', Trans. Amer. Math. Soc. 354 (2002) no. 3, 1027-1047. See also http://arXiv.org, arXiv:math.GT/9912145.
• The postscript version of my thesis in its original form, with all the official front matter and so on. The title is: Symplectic 4-Dimensional 2-Handles and Contact Surgery  along Transverse Knots.

• An expository paper: Why $G(4,2)$ is $S^2 \times S^2$, with A. Lewis, 1996. PDF version
• Another expository paper, written for a Riemannian geometry course under Alan Weinstein at U.C. Berkeley: 4-Manifolds which are Homeomorphic but not Diffeomorphic, 1995. PDF version