Abstract. In this note, we introduce a relative (or Lagrangian) version of the Seidel homomorphism that assigns to each homotopy class of paths in Ham(M ), starting at the identity and ending on the subgroup that preserves a given Lagrangian submanifold L, an element in the Floer homology of L. We show that these elements are related to the absolute Seidel elements by the Albers map. We also study, for later use, the effect of reversing the signs of the symplectic structure as well as the orientations of the generators and of the operations on the Floer homologies.

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism. 53D12, 53D40; 53C15, 53D45, 57R58, 57S05, 58B20

Abstract. We present a deRham model for Chen-Ruan cohomology ring of abelian orbifolds. We introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudo-holomorphic orbifold curves. Thus our model can be viewed as the classical description of Chen-Ruan cohomology for abelian orbifolds. The model simplifies computation of Chen-Ruan cohomology ring. Using our model, we give a version of wall crossing formula.

We provide a construction of examples of semistable degeneration via toric geometry. The applications include a higher dimensional generalization of classical degeneration of K3 surface into 4 rational components, an algebraic geometric version of decomposing K3 as the fiber sum of two E(1)'s as well as it's higher dimensional generalizations, and many other new examples.

Abstract. We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set,à la Atiyah-Bott.