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Explore advanced concepts in algebraic geometry through this comprehensive lecture on singular supports of constructible sheaves in both equal and mixed characteristics. Learn Beilinson's definition of singular support as a closed conical subset on the cotangent bundle of smooth schemes over fields, and understand the fundamental properties proven through Radon transform techniques. Examine an alternative formulation using Braverman-Gaitsgory's interpretation of local acyclicity, and review Beilinson's existence proof methodology. Discover the challenges and developments in mixed characteristic theory, including the introduction of the Frobenius-Witt cotangent bundle as a replacement for the traditional cotangent bundle in characteristic p fibers. Investigate the definition of singular support and its relative variants in this context, and understand how Beilinson's Radon transform arguments provide proof for the existence of saturation of relative variants. This third installment of a four-part series delivered by Takeshi Saito from the University of Tokyo provides deep insights into cutting-edge research in arithmetic geometry and sheaf theory.
