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We give a brief introduction to virtual fundamental classes, which play an important role in Gromov-Witten theory. We then discuss virtual versions of the equivariant Grothendieck-Riemann-Roch theorem and the non-abelian Atiyah-Bott localization theorem for the Borel-style equivariant Chow groups defined by Totaro-Edidin-Graham. This is a report on joint work with Bhamidi Sreedhar.

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