MSRI Workshop on Symplectic Geometry and Mathematical Physics
March 22-26, 2004

[Back]


Abstracts

D. Joyce (Oxford):  Abelian categories and stability conditions

Let A be an abelian category, such as the category of coherent sheaves on a complex projective variety. I develop the notion of configuration, a finite collection of objects and morphisms in A, indexed by a finite poset (I,<), satisfying some axioms. It is an algebraic tool designed for studying the subobjects of an object in A. Given a stability condition S on A, such as slope stability or Gieseker stability, we can form moduli spaces of configurations with certain of the objects (semi)stable. I define invariants of (A,S) by taking the Euler characteristic of these moduli spaces. These invariants satisfy some universal identities. If S,S' are two slope functions on A we can prove transformation laws between invariants for (A,S) and (A,S'). These laws encode how moduli spaces of (semi)stable objects change as we vary the slope function S. There will also be a triangulated category version of this whole story. I believe these invariants will eventually be important in Homological Mirror Symmetry. Applied to the abelian category of coherent sheaves on a Calabi-Yau m-fold, or its derived category, the theory gives a large system of invariants of the Calabi-Yau m-fold, with identities and transformation laws; essentially these invariants "count" arrangements of (semi)stable coherent sheaves. They, or some generalization, should be an extension of Gromov-Witten invariants, but I don't understand this yet. One should also be able to do the same thing for the Fukaya category of the mirror, where the invariants will "count" arrangements of intersecting special Lagrangians.

D. McDuff (SUNY Stony Brook):  Extensions of the Hamiltonian group

We define a disconnected extension of the Hamiltonian group that detects when the fiberwise symplectic form on a symplectic bundle has a closed extension.

W.-D. Ruan (UIC):  Degeneration of Kahler-Einstein manifolds and minimal Lagrangian (coisotropic) vanishing cycles

We will discuss the relation of differential geometric convergence of Kahler-Einstein manifolds (with negative first Chern class) in the sense of Cheeger-Gromov and the algebraic "canonical" degeneration of the underlying family of algebraic manifolds. The vanishing cycles of the algebraic degeneration can be represented geometrically by "minimal" submanifolds that include (minimal Lagrangian submanifolds, H-minimal Lagrangian and coisotropic submanifolds). As application, we will discuss amoeba-type degeneration of Kahler-Einstein hypersurfaces in complex torus into generalized pair of pants.

Y. Ruan (U. of Wisconsin):  Recent advances in orbifold theory

Several years ago, there was a surge of activities to study "stringy" properties of orbifold. After initial phase of explosive growth, main focus now is on the applications. Recently, there are some very exciting developments on this front. They involves some classical questions in algebraic geometry, representation theory and quantum cohomology. Many of them are still on going right now. In this talk, we will survey some of developments.