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Complex K-theory is a generalized cohomology theory introduced by Atiyah and Hirzebruch, which associates to each finite cell complex X the Grothendieck group KU(X) of complex vector bundles on X. However, it also admits a purely algebraic description which makes no mention of vector bundles: it is the complex-oriented cohomology theory associated to the multiplicative formal group over Spec(Z). In this talk, I'll discuss a variant of this algebraic picture which can be used to recover equivariant complex K-theory (as well as equivariant elliptic cohomology), and explain its relationship with the classical character theory of finite groups and various generalizations thereof.

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