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Explore a one-hour lecture from the Joint IAS/PU Arithmetic Geometry series where Pierre Deligne from the Institute for Advanced Study delves into new tensor categories, specifically examining the groundbreaking work of N. Harman, S. Kriz, A. Snowden, and N. Snyder. Discover how pretannakian categories, which are k-linear abelian categories with finite dimensional Hom groups featuring commutative and associative tensor products, unit objects, and duals, serve as generalizations of linear algebraic groups over k. Learn about the relationship between G and the category Rep(G) of representations of G, and understand how G can be recovered from the category when k is algebraically closed. Examine how recent developments have introduced unexpected and beautiful pretannakian categories, moving beyond the traditional approach of obtaining them through "interpolation" from categories Rep(G) in characteristic 0.
