2010-04-11

Towards an NSF proposal

Well, I had plans to dissect those pictures I posted a week ago, but I'm going to postpone that for now. I need to write an NSF proposal, due some time next week. I'm going to start to draft it here, because I started trying to write it and couldn't think of a good title. Also the NSF website is strangely inaccessible at the moment so I can't waste time fiddling with peripheral stuff like budgets and formatting. Here goes:

My main research topics are:

1. Morse $2$--functions, Cerf theory, and invariants. Here the big dream is to have some unifying picture of Heegaard-Floer-like invariants for smooth manifolds that can be understood as coming from Cerf theory and Morse $2$--function, and to really see that you are counting some obstruction to simplifying some presentation of the manifold. (Just as Morse homology counts an obstruction to simplifying a Morse function or handle decomposition.) There are many subsidiary topics, most not nearly so grand:
   1. Finish the paper with Rob.
   2. Write something about the relationship with open book decompositions. Some interesting related questions are: Can we understand Harer's moves on open book decompositions from the point of view of homotopies of Morse $2$--functions? If two homotopic Morse $2$--functions both have sections, are they homotopic through a homotopy such that at each intermediate time sections exist? How does the homotopy class of the contact structure supported by an open book decomposition relate to the homotopy class of an associated $S^2$--valued Morse $2$--function?
   3. Work on a survey article on Cerf theory with Katrin Wehrheim and Chris Woodward.
   4. What is the proper "combinatorialization" of a Morse $2$--function, in the sense that a handle decomposition (or a handle diagram) is a combinatorialization of a Morse function. In other words, what is the simplest data you can add to a diagram in the base of the singular locus so as to completely determine the total space and the map (up to isotopy)?
   5. Work out something useful to say about the setup in bordered Heegaard-Floer homology; in particular, figure out what the appropriate moves should be on combinatorial descriptions of $4$--dimensional cobordisms between bordered $3$--manifolds.
   6. And then the fanciful stuff, which needs to be said carefully so as not to sound crazy; e.g. might there really be the possibility of finding invariants of homology $4$--spheres? (Homotopy $S^4$'s?!?)
2. My current delayed project with Tom Mark on relating convex neighborhoods of certain configurations of symplectic surfaces to Lefschetz fibrations that contain those configurations as singular fibers.
3. My very delayed project with Olga Buse on contact packing problems and symplectic embeddings of polydisks in symplectic balls. The polydisk embedding question is really whether knottedness can help get better embeddings somehow.
4. The even more delayed collaboration with Margaret Symington on locally toric fibrations on symplectic (and near-symplectic) $4$--manifolds. I'm very interested in whether there is a reasonable notion of locally toric fibrations over non-manifold $2$--complexes with integral affine structures, and whether, allowing for singularities on the boundaries, as we did in our near-symplectic paper, you can get something that is in anyway dual to broken Lefshetz fibrations.