Grigory Garkusha: Linear framed motives in use

This is a joint work with Ivan Panin (St. Petersburg). Many important results for framed motives of algebraic varieties are reduced to linear framed motives. They are heavily used for computing suspension P^1-spectra of smooth algebraic varieties in terms of framed correspondences of Voevodsky. The computation is reminiscent of a theorem by G. Segal for classical spectra. They are also used for computing rational stable motivic homotopy theory. In this talk we shall discuss basic properties of linear framed motives and their applications.