Motivic Stable Homotopy Theory - Lecture 1

Delve into an advanced mathematics lecture on motivic stable homotopy theory presented by Ryomei Iwasa at the Institut des Hautes Etudes Scientifiques (IHES). Explore the expanded framework of motivic stable homotopy developed in collaboration with Toni Annala and Marc Hoyois, which extends beyond Voevodsky's initial theory to encompass non-𝐴1-invariant theories. Examine the connections between this broader approach and algebraic K-theory, as well as p-adic cohomology, including syntomic cohomology. The lecture, spanning 1 hour and 27 minutes, is structured into three main parts: Foundations, introducing the concept of 𝑃1-spectrum as the fundamental framework; Techniques, focusing on P-homotopy invariance for conducting homotopy theory in algebraic geometry while maintaining a non-contractible affine line 𝐴1; and Applications, demonstrating the theory's use in algebraic K-theory of arbitrary qcqs schemes and proving an algebraic analogue of Snaith theorem.