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Explore a comprehensive lecture on mapping spaces and E_2 algebras presented by Vladmir Baranovsky from UC Irvine at the University of Miami. Delve into the intricacies of the Bendersky-Gitler spectral sequence, which converges to the cohomology of mapping spaces Maps(K, Y), where K is a simplicial set and Y is a space. Examine the E_1 term of this spectral sequence, involving chains on configuration spaces of K and cochains on Cartesian powers of Y. Focus on the case where K is a graph or its ribbon thickening, and investigate a conjecture that expresses the E_1 term's differential using standard surjection operations on Y's cochains. Discover an application theorem approximating the cohomology of Maps(K, Y) using the cohomology of Cartesian powers of Y with certain double diagonals removed. Explore potential extensions of these results to E_2 algebras and categories with appropriate structures, involving factorization homology of the 2-dimensional ribbon thickening of K.
