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Explore advanced topology through this comprehensive lecture series delivered during the Junior Trimester Program at the Hausdorff Research Institute for Mathematics. Delve into cutting-edge research across multiple areas of topology including motivic Galois groups, rational homotopy theory, operads, L2-invariants, K-theory, topological Hochschild homology, and fusion systems. Learn from leading experts as they present multi-part lecture sequences on topics such as Joseph Ayoub's three-part series on motivic Galois groups, Benoit Fresse's exploration of rational homotopy theory and little discs operads, Markus Spitzweck's treatment of mixed Tate motives and fundamental groups, and Lars Hesselholt's extensive eight-lecture series on topological Hochschild homology. Examine specialized topics including the Atiyah conjecture, L2-Alexander functions of knots and 3-manifolds, isomorphism conjectures for non-discrete groups, the Farrell-Jones conjecture for mapping class groups, calculus of functors, Higher Grothendieck-Witt groups, and the local structure of finite groups. Gain insights into current research directions through presentations on Brown's dihedral moduli space, higher cyclic operads, finite decomposition complexity, representation rings for fusion systems, and Mackey functors with applications to fusion systems, providing a comprehensive overview of contemporary developments in algebraic and geometric topology.
