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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