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2016　Special lectures on advanced topics in Mathematics N

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Akutagawa Kazuo  Konno Hiroshi 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
MTH.E438
Credits
2
Academic year
2016
Offered quarter
4Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

The main topic of this course is the symplectic quotient, the geometry of quotient space in symplectic geometry. Starting with basic topics such as symplectic manifolds and group operations, the instructor will cover moment mapping and symplectic quotients. Then the instructor will explain toric varieties, an important example of symplectic quotients. In addition the instructor will cover the relationship of symplectic quotients with quotient spaces in algebraic geometry called GIT quotients. Finally the instructor will explain the hyperkähler quotient, an analog in hyperkähler geometry to the symplectic quotient.

The moduli spaces of various subjects of differential geometry are constructed as a symplectic quotient or hyperkähler quotient, and the moduli space of subjects of algebraic geometry as a GIT quotient. In many cases, the symplectic quotient is naturally equated with the GIT quotient. This has led many guiding principles to regular vector bundles, and the existence problem of standard metrics for complex manifolds. With quotient spaces, a variety of properties can be investigated using both geometric and algebraic methods. The goal of this course is to explain the geometry of these sorts of quotient spaces.

Student learning outcomes

・Understand basic properties of symplectic manifolds and group actions
・Understand with basic properties of moment maps and symplectic quotients
・Be familiar with many examples of symplectic toric manifolds
・Understand the relation between symplectic quotients and GIT quotients
・Be familiar with basic examples of hyperkähler quotients

Keywords

symplectic manifold, moment map, symplectic quotient, toric manifold, prequantum line bundle, GIT quotient, hyperkähler quotient, toric hyperkähler manifold

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 symplectic manifolds Details will be provided during each class
Class 2 actions of Lie groups Details will be provided during each class
Class 3 backgrounds from physics Details will be provided during each class
Class 4 moment maps Details will be provided during each class
Class 5 symplectic quotients Details will be provided during each class
Class 6 examples of symplectic quotients Details will be provided during each class
Class 7 symplectic toric manifolds Details will be provided during each class
Class 8 toric manifolds as complex manifolds Details will be provided during each class
Class 9 properties of toric manifolds Details will be provided during each class
Class 10 prequantum line bundles and moment maps Details will be provided during each class
Class 11 quotient spaces in algebraic geometry(GIT quotients) Details will be provided during each class
Class 12 relations between symplectic and GIT quotients Details will be provided during each class
Class 13 hyperkähler quotients Details will be provided during each class
Class 14 toric hyperkähler manifold Details will be provided during each class
Class 15 properties of hyperkähler quotients Details will be provided during each class

Textbook(s)

none required

Reference books, course materials, etc.

none required

Assessment criteria and methods

reports 100%

Related courses

  • ZUA.B301 ： Geometry I
  • MTH.B302 ： Geometry II
  • MTH.B331 ： Geometry III
  • MTH.B341 ： Topology

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

manifold theory, cohomology theory, theory of Lie group

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