Topics in geometry: mirror symmetry

Lectures: 2 sessions / week, 1.5 hours / session

This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor. The main topics will be as follows:

Calabi-Yau manifolds; deformations of complex structures, Hodge theory and periods; pseudoholomorphic curves, Gromov-Witten invariants, quantum cohomology; mirror symmetry at the level of Hodge numbers, Hodge structures, and quantum cohomology.

Coherent sheaves, derived categories; Lagrangian Floer homology and Fukaya categories (in a limited setting); homological mirror symmetry conjecture; example: the elliptic curve.

Special Lagrangian submanifolds and their deformations; Lagrangian fibrations, affine geometry, and tropical geometry; SYZ conjecture: motivation, statement, examples (torus, K3); large complex limits; challenges: instanton corrections.

Matrix factorizations; admissible Lagrangians; examples (An singularities; CP1, CP2); the superpotential as a Floer theoretic obstruction; the case of toric varieties.

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