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Explore an advanced mathematics seminar talk that delves into the Geometric Satake Correspondence, examining the equivalence between symmetric monoidal categories of spherical perverse sheaves on affine Grassmannian and the representation category of Langlands Dual Groups. Begin with an exploration of Affine Grassmannian geometry, focusing on semi-infinite orbits, before progressing to spherical perverse sheaves and demonstrating the faithful and exact nature of the global cohomology functor. Learn about the convolution product on spherical perverse sheaves and its perversity preservation through semi-smallness arguments. Discover the Beilinson-Drinfeld affine Grassmannian and understand how the fusion product's alignment with convolution product creates a commutativity constraint, ultimately establishing the symmetric monoidal nature of the spherical perverse Hecke category and global cohomology as a Tannakian fiber functor.
