Motivic Stable Homotopy Theory - Lecture 3

Delve into an advanced lecture on motivic stable homotopy theory presented by Ryomei Iwasa at the Institut des Hautes Etudes Scientifiques (IHES). Explore the groundbreaking work conducted in collaboration with Toni Annala and Marc Hoyois, which extends Voevodsky's theory to encompass non-𝐴1-invariant theories. Gain insights into the connections between this expanded framework and algebraic K-theory, as well as p-adic cohomology, including syntomic cohomology. The 1-hour and 20-minute presentation is structured into three main parts: Foundations, Techniques, and Applications. Begin by grasping the concept of 𝑃1-spectrum, which forms the cornerstone of motivic stable homotopy theory. Progress to understanding the key technique of P-homotopy invariance, enabling homotopy theory in algebraic geometry while maintaining a non-contractible affine line 𝐴1. Conclude by examining practical applications, focusing on the algebraic K-theory of arbitrary qcqs schemes and the proof of an algebraic analogue to Snaith's theorem, demonstrating how K-theory is derived from the Picard stack by inverting the Bott element.