Math 277 - Section 3 - Topics in Differential Geometry - Fall 2009

D. Auroux - Tue. & Thu., 9:30-11am, Room 3 Evans

"Why can't I see my reflection in the mirror on a television?"
-- Anonymous, on Yahoo! Answers

Note: see also the MIT version of this course.


Lecture notes:

Course outline

This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and/or complex geometry (Math 242 helpful but not required). 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, periods; pseudoholomorphic curves, Gromov-Witten invariants, quantum cohomology; large complex structure limits; mirror symmetry at the level of periods 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); challenges: instanton corrections, ...

4. Beyond the Calabi-Yau case: Landau-Ginzburg models and mirror symmetry for Fanos (if time permits)

Matrix factorizations; admissible Lagrangians; examples (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.