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Explore the mathematical theory of singular supports in this comprehensive lecture that examines Beilinson's foundational work on constructible sheaves over smooth schemes. Learn how singular supports are defined as closed conical subsets on cotangent bundles and discover the fundamental properties established through Radon transform techniques. Delve into the Braverman-Gaitsgory interpretation of local acyclicity as an alternative formulation of these concepts. Examine the challenges and developments in mixed characteristic theory, where traditional cotangent bundles are replaced by Frobenius-Witt cotangent bundles that maintain correct rank while being defined specifically on characteristic p fibers. Understand how to define both singular supports and their relative variants in this context, and see how Beilinson's Radon transform arguments provide proofs for the existence of saturation in relative variants. Master advanced concepts in algebraic geometry and sheaf theory through detailed mathematical exposition covering both equal and mixed characteristic cases.
