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Last updated:

October 15, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

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.

Course Curriculum

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

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Massachusetts Institute of Technology
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