Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Overview
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:

Hodge structures, quantum cohomology, and mirror symmetry
CalabiYau manifolds; deformations of complex structures, Hodge theory and periods; pseudoholomorphic curves, GromovWitten invariants, quantum cohomology; mirror symmetry at the level of Hodge numbers, Hodge structures, and quantum cohomology.

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.

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.

Beyond the CalabiYau case: LandauGinzburg models and mirror symmetry for Fanos
Matrix factorizations; admissible Lagrangians; examples (A_{n} singularities; CP^{1}, CP^{2}); the superpotential as a Floer theoretic obstruction; the case of toric varieties.
Calendar
LEC #  TOPICS 

1  The origins of mirror symmetry; overview of the course 
2  Deformations of complex structures 
3  Deformations continued, Hodge theory; pseudoholomorphic curves, transversality 
4  Pseudoholomorphic curves, compactness, GromovWitten invariants 
5  Quantum cohomology and Yukawa coupling on H^{1,1}; Kähler moduli space 
6  The quintic 3fold and its mirror; complex degenerations and monodromy 
7  Monodromy weight filtration, large complex structure limit, canonical coordinates 
8  Canonical coordinates and mirror symmetry; the holomorphic volume form on the mirror quintic and its periods 
9  PicardFuchs equation and canonical coordinates for the quintic mirror family 
10  Yukawa couplings and numbers of rational curves on the quintic; introduction to homological mirror symmetry 
11  Lagrangian Floer homology 
12  Lagrangian Floer theory: Hamiltonian isotopy invariance, grading, examples 
13  Lagrangian Floer theory: product structures, A_∞ equations 
14  Fukaya categories: first version; Floer homology twisted by flat bundles; defining CF(L,L) 
15  Defining CF(L,L) continued; discs and obstruction. Coherent sheaves, examples, introduction to ext. 
16  Ext groups; motivation for the derived category 
17  The derived category; exact triangles; homs and exts. 
18  Twisted complexes and the derived Fukaya category; Dehn twists, connected sums and exact triangles 
19  Homological mirror symmetry: the elliptic curve; theta functions and Floer products 
20  HMS for the elliptic curve: Massey products; motivation for the SYZ conjecture 
21  The SYZ conjecture; special Lagrangian submanifolds and their deformations 
22  The moduli space of special Lagrangians: affine structures; mirror complex structure and Kähler form 
23  SYZ continued; examples: elliptic curves, K3 surfaces 
24  SYZ from toric degenerations (K3 case); LandauGinzburg models, superpotentials; example: the mirror of CP^{1} 
25  Homological mirror symmetry for CP^{1}: matrix factorizations, admissible Lagrangians, etc. 