Field Arithmetic and the Complexity of Galois Cohomology - Part 1

Explore fundamental concepts of field arithmetic and Galois cohomology in this comprehensive lecture delivered by Daniel Krashen from the University of Pennsylvania at the IAS PCMI Park City Mathematics Institute. Delve into essential topics including definitions and standard interpretations of Galois cohomology classes, connections between algebraic structures and cohomological invariants, and their relationship to motives of projective homogeneous varieties. Learn about the structure of Galois cohomology rings, key open questions concerning period-index, symbol length, and vanishing of Massey products, and examine the relationship between Diophantine and cohomological dimensions. Discover current knowledge and techniques in the field, including applications of Milnor conjectures, quadratic forms, and the development of local-global principles in various contexts. Part of the 2024 Program on Motivic Homotopy Theory, this advanced mathematical discussion requires foundational knowledge in algebraic geometry, algebraic topology, homotopy theory, and familiarity with Galois and étale cohomology.