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Explore an advanced mathematics lecture examining closed string Floer theory and its application to the reconstruction problem in mirror symmetry. Delve into how to reproduce mirror partners of symplectic manifolds using Maslov 0 Lagrangian torus fibration data over base B. Learn about a novel approach that employs a sheaf of affinoid algebras mapping appropriate subsets P to symplectic cohomology, supported on their preimages. Understand the conditions necessary for this sheaf to function as a pushforward of function rings and polyvector fields in mirror dual rigid analytic varieties under affinoid torus fibration. Examine specific applications to almost toric fibrations on K3 surfaces with topological sections, and explore extensions to higher dimensions with Gross-Siebert type singularities. Connect these concepts to a broader framework of homological mirror symmetry using Floer theory with supports, as discussed in U. Varolgunes' related work.
