18.969 | Spring 2009 | Graduate

Topics in Geometry: Mirror Symmetry

Lecture Notes

The lecture notes were prepared by Kartik Venkatram in collaboration with Professor Auroux.

1 The origins of mirror symmetry; overview of the course (PDF)
2 Deformations of complex structures (PDF)
3 Deformations continued, Hodge theory; pseudoholomorphic curves, transversality (PDF)
4 Pseudoholomorphic curves, compactness, Gromov-Witten invariants (PDF)
5 Quantum cohomology and Yukawa coupling on H1,1; Kähler moduli space (PDF)
6 The quintic 3-fold and its mirror; complex degenerations and monodromy (PDF)
7 Monodromy weight filtration, large complex structure limit, canonical coordinates (PDF)
8 Canonical coordinates and mirror symmetry; the holomorphic volume form on the mirror quintic and its periods (PDF)
9 Picard-Fuchs equation and canonical coordinates for the quintic mirror family (PDF)
10 Yukawa couplings and numbers of rational curves on the quintic; introduction to homological mirror symmetry (PDF)
11 Lagrangian Floer homology (PDF)
12 Lagrangian Floer theory: Hamiltonian isotopy invariance, grading, examples (PDF)
13 Lagrangian Floer theory: product structures, A_∞ equations (PDF)
14 Fukaya categories: first version; Floer homology twisted by flat bundles; defining CF(L,L) (PDF)
15 Defining CF(L,L) continued; discs and obstruction. Coherent sheaves, examples, introduction to ext. (PDF)
16 Ext groups; motivation for the derived category (PDF)
17 The derived category; exact triangles; homs and exts. (PDF)
18 Twisted complexes and the derived Fukaya category; Dehn twists, connected sums and exact triangles (PDF)
19 Homological mirror symmetry: the elliptic curve; theta functions and Floer products (PDF)
20 HMS for the elliptic curve: Massey products; motivation for the SYZ conjecture (PDF)
21 The SYZ conjecture; special Lagrangian submanifolds and their deformations (PDF)
22 The moduli space of special Lagrangians: affine structures; mirror complex structure and Kähler form (PDF)
23 SYZ continued; examples: elliptic curves, K3 surfaces (PDF)
24 SYZ from toric degenerations (K3 case); Landau-Ginzburg models, superpotentials; example: the mirror of CP1 (PDF)
25 Homological mirror symmetry for CP1: matrix factorizations, admissible Lagrangians, etc. (PDF)

Course Info

As Taught In
Spring 2009