Title: An Introduction to Framed Correspondences
Abstract: In classical homotopy theory, the category of spectra provides a nice way to study cohomology theories. In a similar vein, motivic homotopy theory is at least in part conceived to study cohomology theories in algebraic geometry. Classically, in some special cases, cohomology theories admit certain covariant functoriality known as transfers. In the early 2000s, Voevodsky introduced the theory of framed correspondences encoding certain transfers in motivic homotopy theory, in hope that it will provide a more computationally tractable characterization of the theory of motivic spectra. This idea has since been developed further by Ananyevskiy, Garkusha, Neshitov, and Panin in terms of framed motives. More recently, using this theory, Elmanto-Hoyois-Khan-Sosnilo-Yakerson proved a recognition principle of motivic inifinte P^1 loop spaces over a perfect field. In this talk, I will introduce the theory of framed correspondences and outline their results. Time permitting, I will also discuss some further applications of the theory.