Introduction to Equivariant K-theory - Part II

This lecture is the second part of a series on Equivariant K-theory, presented by Dave Anderson at the Institute for Advanced Study's Special Year Seminar II. Explore the foundations and applications of equivariant K-theory, which emerged in the 1950s from Grothendieck's formulation of the Riemann-Roch theorem. Learn how this mathematical framework helps calculate spaces of sections of vector bundles on varieties through intersection theory, with special focus on the case where G=T is a torus. Understand the localization theorem, K-theoretic equivariant multiplicity, and applications to flag varieties and toric varieties. Discover how these concepts lead to the Weyl character formula for irreducible representations of semisimple Lie groups and Brion's formula for lattice points in a polytope. The lecture also covers the equivariant Riemann-Roch theorem connecting equivariant K-theory to equivariant cohomology and Chow groups, and concludes with positivity theorems for Schubert decompositions. The 1 hour 53 minute session takes place on March 6, 2025, at 10:00am in Simonyi 101.