Abstract

Many quantities in mathematics can be expressed as the integral of a differential form on a manifold.  For example, by the Gauss--Bonnet theorem the integral of the Gaussian curvature of a compact oriented Riemannian surface is 2π times the Euler characteristic of the surface, a topological invariant.  On the other hand, by the Hopf index theorem, the Euler characteristic is the sum of the indices of the zeros of a continuous vector field on the surface.  Putting the two theorems together, we obtain the integral of the curvature form as a finite sum over the zeros of a vector field.

With this example in mind, one might wonder if there is a general method to convert an integral on a manifold to a sum over a finite set.  The natural context for this problem is when there is a group acting on the manifold with finitely many fixed points, for then one can ask if an integral over the manifold is equal to a sum over the fixed point set.  Remarkably, it turns out that the answer is yes, but only for certain types of differential forms called equivariantly closed forms. The formula was discovered in 1982 by Atiyah and Bott on the one hand and independently by Berline and Vergne on the other.  The equivariant closed forms were introduced by Henri Cartan thirty years earlier to study the cohomology of a space with a group action, called equivariant cohomology.

This formula is of great computational utility and has found applications in topology, symplectic geometry, algebraic geometry, and physics.  The goal of the course is to study the development of these ideas leading to a proof of the localization formula of Atiyah--Bott—Berline--Vergne and to see some of the applications.  We will need to draw on techniques from algebraic topology, differential geometry, Lie groups and Lie algebras, representation theory, and commutative algebra, but I will try to explain the techniques assuming a knowledge of manifolds and algebraic topology.

Reference 

Loring W. Tu, Elements of Equivariant Cohomology, to be published by Springer. Manuscript available from the instructor.

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