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Mirror symmetry shows the relationship between Geometry Objects called Calabi-Yau Manifolds. To express various mathematical versions of mirror symmetry, you need to know the geometric concepts. Focusing on various aspects of mirror symmetry, this [course_title] is suitable for you having knowledge of symplectic and complex geometry. So, this [course_title] you will be able to aptly understand Hodge Structures, Homological Mirror Symmetry, Lagrangian Fibrations, beyond the Calabi-Yau Case etc.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: MIT

Course Curriculum

Module: 01
The origins of mirror symmetry; overview of the course 00:30:00
Deformations of complex structures 00:30:00
Deformations continued, Hodge theory; pseudoholomorphic curves, transversality 00:30:00
Pseudoholomorphic curves, compactness, Gromov-Witten invariants 00:45:00
Quantum cohomology and Yukawa coupling on H1,1; Kähler moduli space 00:45:00
Module: 02
The quintic 3-fold and its mirror; complex degenerations and monodromy 00:45:00
Monodromy weight filtration, large complex structure limit, canonical coordinates 00:30:00
Canonical coordinates and mirror symmetry; the holomorphic volume form on the mirror quintic and its periods 00:45:00
Picard-Fuchs equation and canonical coordinates for the quintic mirror family 00:30:00
Yukawa couplings and numbers of rational curves on the quintic; introduction to homological mirror symmetry 00:45:00
Module: 03
Lagrangian Floer homology 00:30:00
Lagrangian Floer theory: Hamiltonian isotopy invariance, grading, examples 00:30:00
Lagrangian Floer theory: product structures, A_∞ equations 00:30:00
Fukaya categories: first version; Floer homology twisted by flat bundles; defining CF(L,L) 00:45:00
Defining CF(L,L) continued; discs and obstruction. Coherent sheaves, examples, introduction to ext. 00:30:00
Module: 04
Ext groups; motivation for the derived category 00:45:00
The derived category; exact triangles; homs and exts. 00:30:00
Twisted complexes and the derived Fukaya category; Dehn twists, connected sums and exact triangles 00:30:00
Homological mirror symmetry: the elliptic curve; theta functions and Floer products 00:30:00
HMS for the elliptic curve: Massey products; motivation for the SYZ conjecture 00:45:00
Module: 05
The SYZ conjecture; special Lagrangian submanifolds and their deformations 00:30:00
The moduli space of special Lagrangians: affine structures; mirror complex structure and Kähler form 00:45:00
SYZ continued; examples: elliptic curves, K3 surfaces 00:30:00
SYZ from toric degenerations (K3 case); Landau-Ginzburg models, superpotentials; example: the mirror of CP1 00:45:00
Homological mirror symmetry for CP1: matrix factorizations, admissible Lagrangians, etc. 00:30:00
Assessment
Submit Your Assignment 00:00:00
Certification 00:00:00

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