2020 Special lectures on advanced topics in Mathematics D

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Ueda Kazushi  Honda Nobuhiro 
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
Intensive (Zoom)  
Group
-
Course number
MTH.E434
Credits
2
Academic year
2020
Offered quarter
3Q
Syllabus updated
2020/9/23
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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.

Student learning outcomes

・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

Keywords

complex manifold, algebraic variety, symplectic geometry, G-structure, Calabi-Yau manifold, derived category of coherent sheaves, Fukaya category, mirror symmetry

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  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.

Textbook(s)

None required

Reference books, course materials, etc.

Lecture note is available at
https://www.ms.u-tokyo.ac.jp/~kazushi/course/tit2020.pdf

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.E640 : Special lectures on current topics in Mathematics J
  • ZUA.E334 : Special courses on advanced topics in Mathematics D

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Not in particular

Other

Not in particular

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