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Explore advanced topics in arithmetic geometry through this mathematical lecture examining restriction maps in cohomology from smooth projective varieties to affine subvarieties. Delve into Grothendieck's generalized Hodge conjecture and its purely algebraic expressions, focusing on how restriction maps behave when moving from a smooth projective variety X to an affine subset U. Investigate de Rham cohomology over dense sets of closed points in smooth schemes over ℤ, and discover how global differential forms on X control these restrictions. Learn about p-adic analogues using Bhatt-Scholze prismatic cohomology and its relationship to derived p-complete de Rham cohomology through separated quotients. Examine concrete examples that demonstrate the necessity of additional assumptions for lifting cohomological control, and study recent results by Caro-D'Addezio on derived p-complete de Rham cohomology modulo p-torsion. Understand how these techniques extend to prismatic cohomology modulo I-torsion and their implications for étale cohomology of rigid analytic fibers, providing insights into fundamental questions in algebraic geometry and number theory.
