An Update on SYZ Mirror Symmetry and Family Floer Theory

Explore an advanced lecture on the Strominger-Yau-Zaslow (SYZ) approach to homological mirror symmetry and its recent developments. Delve into the construction of mirror symmetry using Lagrangian torus fibrations on the complement of an anticanonical divisor. Examine how the mirror is formed as a moduli space of weakly unobstructed objects in the Fukaya category supported on the fibers. Investigate the challenges posed by holomorphic discs with negative Maslov index and their impact on mirror geometry beyond traditional Landau-Ginzburg models. Learn about a proposed Morse-theoretic construction of the Fukaya-Floer algebra for a family of Lagrangian tori, which yields a deformed Cech model for the algebra of polyvector fields on the mirror. Discover how this approach establishes a functor from Lagrangian sections of the SYZ fibration to modules over the constructed algebra. Presented by Denis Auroux from Harvard University, this hour-long lecture offers a deep dive into cutting-edge research in algebraic geometry and symplectic topology.