Introduction to Symplectic Geometry (2018 Spring)

Room: 綜三734
Time: Thursday 456 (11:10-12:30, 1:50-3:00)
Lecture Series on Lagrangian Floer theory by Prof. Ono at NCTS

Course Description

This is an introductory course on symplectic geometry. The material that will be covered in the course includes the following:

Linear symplectic algebra
Symplectic manifolds
Normal form theorems
Lie groups, Hamiltonian group actions
Moment maps, Symplectic reductions
Convexity results
Yang-Mills theory (if time permits)

References

A. Silva, Lectures on Symplectic Geometry
L. Jeffrey, Hamiltonian Group Actions and Symplectic Reduction, IAS/Park City Mathematics Series
E. Meinrenken, Lecture notes on Symplectic Geometry
V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (especially Chap 1 for Physical motivation)
M. Audin, The Topology of Torus Actions on Symplectic Manifolds
D. McDuff, D. Salamon, Introduction to Symplectic Topology

Background:

L.Tu, An Introduction to Manifolds
L. Duistermaat, Lie groups (1.1-1.4: Lie group and Lie algebra; 2.1-2.4: group actions and associate bundles)
S. Kobayashi, Foundations of Differential Geometry I & II

Syllabus

2/22: Motivation, symplectic vector space, symplectic linear reduction
3/1: Move to 6/7 and 6/14
3/8: Compatible complex structure, linear symplectomorphism
3/15: Symplectic manifold: cotangent bundle, Poisson bracket (assuming knowledge of differential forms and vector fields on manifolds)
3/22: Almost complex structure, Integrability
3/29: Integrability, submanifolds
4/5: National Holiday
4/12: Normal forms, Lie group action

4/19: Hamiltonian action
4/26: Symplectic reduction
5/3: Various examples including exact manifolds
5/10: More on moment maps, reduction in stages
5/17: Morse theory and convexity
5/24: Convexity and toric manifold
5/31: Toric manifold and D-H formula
6/7: 10:30-3:20
6/14: starts at 10:30, no precise ends

Evaluation

Presentations (either mathematics or physics). Presentation ordering: 施/ 堯/ 賴/晉/官/ 胡/ 俊/ 鳴/庭

Last Updated: May 21, 2018
URL: http://www.math.nthu.edu.tw/~nankuo/ISG.html