The lecture notes were prepared by Kartik Venkatram in collaboration with Professor Auroux.
| LEC # | TOPICS | LECTURE NOTES |
|---|---|---|
| 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) |