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Explore the P^1-motivic homotopy theory and its applications in a 58-minute lecture by Longke Tang at the Hausdorff Center for Mathematics. Delve into this generalization of A^1-motivic homotopy theory, which relaxes the requirement of A^1 contractibility and focuses on the invertibility of pointed P^1. Discover how this approach extends to cohomology theories with nontrivial reduced cohomology of A^1, such as Hodge cohomology, de Rham cohomology, and prismatic cohomology. Learn about the construction of the P^1-motivic cycle map and its role in providing a unified framework for cycle maps in various cohomology theories. If time allows, examine the application of this cycle map in proving prismatic Poincaré duality.
