2007-10-26

Rebecca Goldin, George Mason University and Cornell University

Schubert Calculus and Equivariant Cohomology

Schubert calculus began at the end of the nineteenth century as a set of questions in enumerative geometry. For example, given four generic lines in three-dimensional space, how many other lines intersect all four? This question and generalizations can be reformulated in terms of the intersection of certain varieties in G/B, where G is a complex reductive group and B is a Borel subgroup. In this talk, we explain the basic set up in Schubert calculus, and pose an open question of "positivity"of structure constants. We then explore some methods for computing these numbers in equivariant cohomology. We end up with an operator in the Demazure algebra that, when applied to "1" yields a formula for Schubert structure constants. This is joint work with Allen Knutson, Univ. of CA, San Diego.