Figure 1: Two families of ellipses parameterized by a circle. The first family is trivial and all the ellipses in it are positioned in exactly the same way. The ellipses in the second family are all the same but are rotated in their planes. Families of this kind are called isotrivial. The usual parameter spaces do not capture the distinction between trivial and isotrivial families.
Another common use of stacks is for the study of homological properties of quotient singularities in topology and geometry. Any singular space which is locally obtained as a quotientt of a manifold by a finite group of symmetries can be effectively replaced by a special kind of a smooth stack called an orbifold. The orbifolds are stacks that look generically like spaces but have non-trivial groups of symmetries attached to some special points. Smooth orbifolds have been successfully studied with the usual tools of differential geometry and topology and have lead to powerful results about singular spaces.
Figure 2:The complex quadratic cone z2=xy can be viewed as the quotient of the complex plane by the central symmetry with respect to the origin. The vertex of the cone is an orbifold point. The orbifold structure ``remembers'' the involution used to create the cone. The vertex is a singular point of the underlying space (the cone) but is a smooth point of the stack (the orbifold).
Today, with the the development of the non-abelian Hodge theory and the flourishing interaction between Algebaric Geometry and String Theory, the algebraic (n-)stacks are clearly outgrowing their basic function as a convenient framework and language for studying moduli and are becoming an essential tool in everyday research. The Algebraic Stacks program at MSRI will summarize, centralize and disseminate these very important new directions.
The program will be organized around the following topics:
- Intersection theory on stacks and virtual fundamental cycles.
- Non-abelian Hodge theory and geometric n-stacks.
- Perverse sheaves on stacks and the geometric Langlands program.
- D-brane charges in string theory and characteristic classes of sheaves on stacks.
- Moduli of gerbes and mirror symmetry.
|January 10, 2002 - January 16, 2002||Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory|
|March 11, 2002 - March 15, 2002||Intersection Theory on Stacks|
|March 28, 2002 - April 05, 2002||Non-Abelian Hodge Theory|