# Mathematical Sciences Research Institute

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# Program

Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory January 07, 2002 to May 17, 2002
Organizers W. Fulton, L. Katzarkov, M. Kontsevich, Y. Manin,R. Pandharipande, T. Pantev, C. Simpson and A. Vistoli
Description This program is concerned with the notion of a stack which is used to parameterize geometric objects varying in families. Such families are classically known as moduli and their understanding is a central theme in algebraic geometry. The formalism of algebraic stacks was introduced by A.Grothendieck and M.Artin as the natural context in which moduli problems of objects with symmetries can be tackled. The usual spaces used in geometry - manifolds, varieties, schemes - are inadequate for the parameterization of geometric objects that are self-similar. Instead one needs a procedure that will not only encode the way things vary in families but will also remember the intrinsic symmetries of each object in the family.

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.
The broad span of each of these areas clearly indicates that the field of algebraic stacks has gathered a huge momentum and is bound to become one of the main tools of the working mathematician. The fact that the subject relates to quantum cohomology and mirror symmetry, modern theoretical physics, geometric Langlands program and geometric representation theory, modern homotopy theory and symplectic geometry makes it a desirable subject to learn for many young PhD's and graduate students in Mathematics and Theoretical Physics.

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Programmatic Workshops
 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