Abstract:
Holomorphic 2-forms on K\"{a}hler surfaces lead to "Local Gromov-Witten invariants" of spin curves. This paper shows how to derive sum formulas for such local GW invariants from the sum formula for GW invariants of certain ruled surfaces. These sum formulas also verify the Maulik-Pandharipande formulas that were recently proved by Kiem and Li.

Abstract:
Recently, Gunningham \cite{G} calculated all spin Hurwitz numbers in terms of combinatorics of Sergeev algebra. In this paper, we use a spin curve degeneration to obtain a recursion formula for degree three spin Hurwitz numbers.

Abstract:
We use simple geometric arguments to calculate the dimension zero local Gromov-Witten invariants of elliptic multiple fibers. This completes the calculation of all dimension zero GW invariants of elliptic surfaces with $p_g>0$.

Abstract:
We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from \cite{ip3}. In particular, as in \cite{bl1}, these calculations also confirm the well-known Yau-Zaslow Conjecture \cite{yz} for primitive classes in $K3$ surfaces.

Abstract:
The usual Gromov-Witten invariants are zero for K\"{a}hler surfaces with $p_g\geq 1$. In this paper we use analytic methods to define Family Gromov-Witten Invariants for K\"{a}hler surfaces. We prove that these are well-defined invariants of the deformation class of the K\"{a}hler structure.

Abstract:
On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form $\a$, there is an almost complex structure associated with $\a$. We show how this implies vanishing theorems for the Gromov-Witten invariants of $X$. This extends the approach, used in \cite{lp} for K\"{a}hler surfaces, to higher dimensions.

Abstract:
Gunningham [G] constructed an extended topological quantum field theory (TQFT) to obtain a closed formula for all spin Hurwitz numbers. In this note, we use the gluing theorem in [LP2] to reprove the Gunningham's formula. We also describe a TQFT formalism naturally induced from the gluing theorem.

Abstract:
The query for triple information on product–attribute (property)–value is one of the most frequent queries in e-commerce. In storing the triple (product–attribute–value) information, a vertical schema is effective for avoiding sparse data and schema evolution, while a conventional horizontal schema often shows better query performance, since the properties are queried as groups clustered by each product. Therefore, we propose two storage schemas: a vertical schema as a primary table structure for the triple information in RDBMS and a pivoted table index created from the basic vertical table as an additional index structure for accelerating query processing. The pivoted table index is beneficial to improving the performance of the frequent pattern query on the group properties associated with each product class.

Abstract:
The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These "spin Hurwitz numbers", recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors' previous work, they are also related to the Gromov-Witten invariants of Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spectral flow of certain anti-linear deformations of the d-bar operator, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs.

Abstract:
This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants vanish for most homology classes. This involves extending the Splitting Formula and the Symplectic Sum Formula to the family GW invariants introduced by the first author. We present applications to the invariants of elliptic surfaces and to the Yau-Zaslow Conjecture. In both cases the results agree with the conjectures of algebraic geometers and yield a proof, to appear in [LL1], of previously unproved cases of the Yau-Zaslow Conjecture.