18.969 - Topics in Geometry - Spring 2009

D. Auroux - Tuesdays & Thursdays, 11-12:30 in 2-142.

Lecture notes:

There are two sets of notes: handwritten notes by the lecturer, and typeset notes taken by Kartik Venkatram. Both are provided as is, without any guarantee of readability or accuracy.

Course description

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:

1. Hodge structures, quantum cohomology, and mirror symmetry

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.

2. A brief overview of homological mirror symmetry

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

3. Lagrangian fibrations and the SYZ conjecture

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, ...

4. Beyond the Calabi-Yau case: Landau-Ginzburg models and mirror symmetry for Fanos

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


This very incomplete list tries to provide some of the more accessible references on the material. There are many other excellent references, but those often require a higher level of dedication.