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Research

Papers

• My papers are available for download via the ArXiv here.
• Bibliographic information can be found at MathSciNet (reviews will appear in due course).
• Here is an individual listing:
  
  

  Title	ArXiv	Publication information
  Arithmetic mirror symmetry for the 2-torus (with Y. Lekili)	1211.4632	submitted to Memoirs of the AMS
  Current working draft: Aug. 2015 (see status update below).
  Fukaya categories of the torus and Dehn surgery (with Y. Lekili)	1102.3160	Proc. Natl. Acad. Sci. USA 108 (2011), no. 20, 8106-8113.
  A symplectic Gysin sequence	0807.1863	Preprint
  Hamiltonian handleslides for Heegaard Floer homology	0801.0564	Proc. 14th Gokova Geometry-Topology Conference
  Symplectic fibrations and the abelian vortex equations	math/0606063	Comm. Math. Phys. (2007). The original publication is available at www.springerlink.com.
  Lagrangian matching invariants for fibred four-manifolds: II	math/0606062	Geometry & Topology 12 (2008) The published article is available here.
  Lagrangian matching invariants for fibred four-manifolds: I	math/0606062	Geometry & Topology 11 (2007). The published article is available here.
  Zero-sets of near-symplectic forms	math/0601320	J. Symp. Geom. 4 (2006). The published article is available here.
  A remark on Kähler forms on symmetric products of Riemann surfaces	math/0501547	Unpublished note, now subsumed by Hamiltonian handleslides, listed above.
  Surface-fibrations, four-manifolds, and symplectic Floer homology	-	Ph.D. thesis, Imperial College London, 2005. The PDF is here.


Status update on Arithmetic mirror symmetry for the 2-torus (Aug. 2015). This paper is undergoing significant revision and expansion to address concerns (identified by the referees) about the meaning of quasi-equivalences of A-infinity categories over non-fields. The revision also eliminates an ad hoc lattice counting argument, replacing it by a much better one. The draft here is missing an appendix about invariance of Fukaya categories over non-fields, and needs a little more work in the main text, but is nearly complete. We expect to resubmit it to Memoirs shortly.

Core homological mirror symmetry project

• Curve counts from categories in Calabi-Yau mirror symmetry.
  A very long research statement describing joint work with Sheridan and with Ganatra/Sheridan (errors are mine). Document.
• (with N. Sheridan) Automatic split-generation of Fukaya categories via mirror symmetry. This version Oct. 1 is complete; Arxiv submission imminent. Draft
  Proves that "core homological mirror symmetry" implies homological mirror symmetry, modulo symplectic geometry foundations.
• (with N. Sheridan) Constructing the relative Fukaya category In preparation, expected completion Fall 2015. Draft
  Symplectic geometry foundations for "automatic split-generation" paper. Sets up the Fukaya category of a CY manifold relative to an ample divisor, along with some related structures including the closed-open string map.
• (with S. Ganatra and N. Sheridan) From categories to curve-counts, via mirror symmetry In preparation, nearly complete. Expected completion: Sept. 2015. Draft
  Shows how core HMS implies Hodge-theoretic mirror symmetry. Depends on two foundational papers: one by Sheridan on homological algebra, and another crucial one on symplectic geometry, listed below.
• (with S. Ganatra and N. Sheridan) Fukaya categories and semi-infinite variations of Hodge structure In preparation. Expected completion: Fall. 2015.
  Shows that the cyclic open-closed map, from the cyclic homology of the relative Fukaya category to quantum cohomology, respects natural connections and pairings. This paper is not yet available, even in draft form. However, the summary document, listed above, gives a detailed summary of the statements and proofs.

Outline

This is a technical summary aimed at other mathematicians, and full of geometers' jargon. I hope soon to complement it with a lay-person's account.

My work has two closely-related aspects; in fact a central theme is the interplay between them. The first aspect concerns four-manifolds and TQFTs in 3+1 dimensions. I am interested particularly in near-symplectic geometry and in maps from 4-manifolds to surfaces (Lefschetz fibrations and their generalisations).

The second aspect concerns symplectic geometry, particularly symplectic Floer homology. I am interested in the structures in Floer homology arising from Lagrangian correspondences between symplectic manifolds.

The two aspects come together by means of a sort of topological field theory for 3- and 4-manifolds singularly fibred by surfaces, based on the idea that Lagrangian correspondences between symplectic manifolds (in this case, symmetric products of Riemann surfaces) can serve as boundary conditions for pseudo-holomorphic curves. This theory, of what I call Lagrangian matching invariants, is an example of a much more general formalism in symplectic topology. Two pressing questions about it are to show that it gives invariants of 4-manifolds, independent of the fibrations on them, and to show that it recovers Seiberg-Witten theory. Neither of these questions is close to being answered, but investigating them has led me to study some new structures in symplectic geometry which hold independent interest.

This project has grown out of my Ph.D. thesis (Imperial College London, 2005) and out of my attempt to answer a wonderful problem posed by my supervisor, Simon Donaldson: that of describing in symplectic terms the Seiberg-Witten invariants for a four-manifold equipped with a "singular Lefschetz pencil" (or "broken pencil").

Reviews

I review papers for Mathematical Reviews. If your institution subscribes to MathSciNet you can view them here.

Talks

Lagrangian correspondences and three-manifold invariants, MSRI, March 2010.

I have given several variants of a talk entitled Broken pencils and four-manifold invariants. This one was given at Durham University in 2007.

A lecture given in Harvard in February 2007 entitled Fibred 3-manifolds and the Floer homology of fibred Dehn twists.

Two contributions to a reading seminar organised by Yann Rollin at Imperial College in 2006. The topic was Kronheimer and Mrowka's monopole Floer homology theory for 3-and 4-manifolds, set out in their book.

• Eigenspinors and the Chern-Simons-Dirac functional.
• Flat connections, Dirac operators and the triple-product form.