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Explore the foundations and applications of motivic homotopy theory through this comprehensive graduate summer school program from the Park City Mathematics Institute. Delve into this powerful mathematical framework that emerged from Morel and Voevodsky's groundbreaking work in the 1990s, which has since become an essential tool for understanding arithmetic aspects in algebra and algebraic geometry while serving as a fascinating generalization of classical homotopy theory. Master key concepts through interwoven minicourses covering unstable motivic homotopy theory, characteristic classes in stable motivic homotopy theory, motivic applications in enumerative geometry, and motivic versions of the Weil conjectures. Study advanced topics including A¹-algebraic topology, Chow groups, algebraic vector bundles over smooth affine varieties, torsors over affine curves, and Massey products in Galois cohomology. Examine the arithmetic properties of local systems, fundamental problems in Galois cohomology of fields, and aspects of G-bundles in algebraic geometry. Learn from leading experts including Joseph Ayoub, Kirsten Wickelgren, Philippe Gille, Michael Hopkins, Burt Totaro, Fabien Morel, Hélène Esnault, Aravind Asok, and others who combine mathematical expertise with exceptional teaching abilities. Participate in daily problem sessions designed to develop practical facility with complex theoretical material. Prerequisites include basic knowledge of algebraic geometry, algebraic topology, and homotopy theory, with familiarity in Galois cohomology and étale cohomology beneficial for certain courses.
