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Explore a detailed lecture on homological mirror symmetry (HMS) for symplectic Calabi-Yau manifolds, focusing on methods to access derived Fukaya categories through mirror symmetry. Learn about the emerging proofs of HMS in complete intersection CY's within toric varieties, with particular emphasis on Seidel's technique of proving HMS in large volume limits. Discover intrinsic mirror construction methods like the Gross-Siebert program and Fukaya's Family Floer theory, which incorporate quantum corrections directly. Examine a new approach to mirror symmetry being developed by Abouzaid, Groman, and Varolgunes, potentially applicable to proving HMS for intrinsically constructed mirrors. Delve into the mathematical framework that connects derived categories of coherent sheaves with their mirror counterparts in projective varieties over formal Laurent series fields.
