Topics in Geometry: Mirror Symmetry

Syllabus

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:

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 (A_n singularities; CP¹, CP²); 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, Gromov-Witten invariants
5	Quantum cohomology and Yukawa coupling on H1,1; Kähler moduli space
6	The quintic 3-fold 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	Picard-Fuchs 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); Landau-Ginzburg models, superpotentials; example: the mirror of CP1
25	Homological mirror symmetry for CP1: matrix factorizations, admissible Lagrangians, etc.