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Explore advanced mathematical concepts in this joint IAS/PU Symplectic Geometry Seminar lecture delivered by Roman Krutowski from UCLA. Delve into the intricate relationship between Hecke algebras and Morse theory applied to loop spaces, with a particular focus on Higher-dimensional Heegaard Floer homology (HDHF). Learn how HDHF extends Lipshitz's cylindrical reformulation of Heegaard Floer homology from surfaces to arbitrary Liouville domains, and discover its role as a model for Lagrangian Floer homology of symmetric products. Examine a Morse-theoretic model that enables computations of the HDHF A∞-algebra of k cotangent fibers in the cotangent bundle of smooth manifolds. Follow the explicit computation of this A∞-algebra for the cotangent bundle of the 2-dimensional sphere, which yields a differential graded algebra that can be interpreted as the derived HOMFLYPT skein algebra of the sphere. This presentation bridges symplectic geometry, algebraic topology, and knot theory, offering insights into cutting-edge research in mathematical physics and geometry.
