We will start with recalling the definitions of almost complex structures and their integrability, and then discuss the functor of points approach to algebraic varieties. Then we will discuss rudiments of symplectic geometry, and compare complex and symplectic geometry using the language of G-structures. After brief introductions to derived categories of coherent sheaves and Fukaya categories, we will discuss several aspects of mirror symmetry. If the time permits, we will also discuss special Lagrangian torus fibrations and tropical geometry.
Mirror symmetry is a mysterious relationship, originally suggested by string theorists, between complex geometry of one space and symplectic geometry of another space, called the mirror of the original space. The aim of this lecture is to discuss similarities and differences between complex and symplectic geometry, and give a gentle introduction to mirror symmetry.
・Definitions of complex manifolds and algebraic varieties
・Basic definitions and results in symplectic geometry, and their motivations from classical mechanics
・Relations among complex, symplectic, and Riemannian geometry from the point of view of G-structures
・Concrete examples of derived categories of coherent sheaves and Fukaya categories
・Some familiarity with mirror symmetry
complex manifold, algebraic variety, symplectic geometry, G-structure, Calabi-Yau manifold, derived category of coherent sheaves, Fukaya category, mirror symmetry
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | The following topics will be covered in this order : -- definitions and examples of complex manifolds and algebraic varieties -- definitions and examples of symplectic manifolds -- Hamilton's equation of motion, Noether's theorem, Liouville-Arnold theorem -- G-structures and their integrability -- definitions and examples of Kaehler manifolds and Calabi-Yau manifolds -- derived categories of coherent sheaves -- Lagrangian intersection Floer theory and Fukaya categories -- mirror symmetry | Details will be provided during each class session. |
None required.
Lecture note is available at
https://www.ms.u-tokyo.ac.jp/~kazushi/course/tit2020.pdf
Assignments (100%).
None in particular