There are two main problems suggested for further study in this proposal. The first problem is the study of the Heegaard Floer homology of a three-manifold obtained by gluing two other three-manifolds with torus boundaries. The gluing is determined once a framing on the boundary of each component is fixed. The principal investigator is interested in constructing a relative Heegaard Floer theory for three-manifolds with framed torus boundary, such that the Floer theory of the resulting closed three-manifold may be computed from the relative Floer theories of the two pieces. Potential applications include an understanding of the Floer theory of satellite knots, and some lower bounds on the slice genus of such knots, at least in the case of the Whitehead double. The second problem deals with the construction of integer valued enumerative invariants of Calabi-Yau threefolds. In an earlier work, the principal investigator has constructed such an invariant for any given homology class which is either primitive or a prime multiple of a primitive class. It counts embedded pseudo-holomorphic curves of a given genus in the Calabi-Yau threefold, which represent the given homology class. The goal is to overcome the technical difficulties, which force the restriction on the homology class, and extend the construction to all homology classes. The principal investigator is also interested in deriving a generating function formula, which relates these invariants to Gromov-Witten invariants, similar to the work of Gopakumar and Vafa in string theory literature.
Counting holomorphic curves in complex or symplectic manifolds has been the source of great developments in topology, in algebraic geometry and in mathematics of string theory, for the past two decades. These counts reveal, as numerical characteristics of the subject of study, very important geometric and topologic features that have been almost impossible to understand before the introduction of these techniques. Floer homology in general, and Heegaard Floer homology in the study of low dimensional manifolds in particular, give examples of revolutionary technologies in this category. Gromov-Witten invariants, providing a language for enumerative geometry, form another example. However, the actual computations are generally very hard. The two problems suggested in this proposal are aimed at a reduction of the computations to potentially simpler cases: For a three dimensional manifold obtained from gluing two simpler three-dimensional pieces we would like to reduce the computation of Heegaard Floer homology to a computation for each piece, and for Gromov-Witten invariants, which are generally rational numbers with crazy denominators, we would like to describe them in terms of integer valued enumerations of potentially simpler objects. Once these techniques are developed, the computations will be possible at least in certain interesting classes of examples. We hope for some fantastic applications of the first problem to the study of knots, and a very interesting link between the construction of the second problem and the counts of BPS-states in physics.