From Schubert calculus to equivariant cohomology
The aim of this series of lectures is to sketch how the enumerative geometry of the 19th century has grown into the thriving field of equivariant cohomology in the 20th and 21st centuries.
We will begin by sketching the history of the subject, starting from Steiner’s question (posed in 1848): how many conics are tangent to 5 given plane conics? Answering such questions became a thriving business in the next half century, with a powerful but unjustified method devised primarily by Schubert. Hilbert's 15th problem asked for a justification. We will discuss how the development of algebraic topology can be used to solve such problems: Schubert's calculus takes place in the cohomology rings of appropriate moduli spaces.
Equivariant cohomology grew out of a seminar led by Borel in 1958-59. When a group acts on a space, there is a richer cohomology theory, now called equivariant cohomology, that takes this action into account. Most of the moduli spaces involved in classical enumerative geometry problems do have such group actions.
We will describe some of what is known (and not known) about equivariant cohomology of algebraic varieties, mostly by example. In particular, we will look at Grassmannians, flag varieties, and toric varieties.