Torsors Over Affine Curves - Part 1

Explore a comprehensive lecture on G-torsors and G-bundles in algebraic geometry through this one-hour presentation from the IAS Park City Mathematics Institute. Delve into the theory of vector bundles and quadratic bundles (Grothendieck-Serre, 1958), focusing specifically on affine smooth connected curves over algebraically closed fields k. Learn how G-torsors become trivial for semisimple k-groups, and examine cases involving Dedekind rings, including the ring of integers Z and affine curves over arbitrary fields. Study important examples like the affine line (Raghunathan-Ramanathan, 1984) and punctured affine line while investigating étale cohomology and patching techniques. Access supplementary materials including detailed lecture notes and problem sets to reinforce understanding of these advanced mathematical concepts. Part of the 2024 Program on Motivic Homotopy Theory, this lecture requires foundational knowledge in algebraic geometry, algebraic topology, and homotopy theory, with additional background in Galois cohomology and étale cohomology being beneficial.