Grigory Garkusha: Algebraic Kasparov K-theory...

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

Title: Grigory Garkusha: Algebraic Kasparov K-theory, framed correspondences and stable motivic homotopy theory (after Cuntz and Voevodsky)

Algebraic Kasparov K-theory is a stable motivic homotopy theory for algebras. A major computation here is the computation of the suspension spectrum of an algebra as an explicit Ω-spectrum. It is based on Cuntz's treatment of Kasparov K-theory (resp. bivariant K-theory of locally convex algebras) as well as on further extension of Cuntz's theory to all algebras by Cortinas-Thom. A computation of the suspension spectrum of a smooth algebraic variety by the speaker and Panin is based on the theory of framed correspondences of Voevodsky. In this talk we will show that both computations, based on Cuntz's and Voevodsky's theories respectively, share lots of common properties. They allow to compute explicitly stable motivic homotopy types of spectra.