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Explore a comprehensive mathematics lecture on equivariant cohomology presented by William Graham at the Institute for Advanced Study. Delve into the foundations of this mathematical concept introduced by Borel in the 1960s, with particular emphasis on torus-equivariant cohomology, Borel-Moore homology, and Chow groups. Learn about flag varieties and Schubert varieties through detailed examples, and understand localization theorems for torus applications that connect equivariant theories to fixed point locus. Master the concept of equivariant multiplicity at torus-fixed points and discover the integration formula applicable to singular varieties. Examine the equivariant cohomology of flag variety through convolution, explore divided difference operators for obtaining Schubert class representatives, and study Bott-Samelson resolutions for equivariant multiplicity formulas. Investigate the relationship between equivariant cohomology and various mathematical concepts including multiplicities, smoothness, and rational smoothness, with potential coverage of weighted flag varieties from joint research with Scott Larson.
