Introduction to Symplectic Geometry (2026 Spring)

Room: 綜三 631
Time: W89 R34 (課程時間細節請注意下方描述)
進度以*標記預估進度，完成的會將*移除。考試日期與範圍是預估，可能會有改變，請同學到時候注意網頁的UPDATE

Course Description

This is an introductory course on symplectic geometry. It assumes the basic knowledge of differential geometry. The material that might be covered in the course includes the following (will be updated throughout the semester):

Linear symplectic algebra
Symplectic manifolds
Lie groups, Hamiltonian group actions
Moment maps, Symplectic reductions
Convexity results
Toric manifolds and Delzant's Theorem

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
V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian T^n-spaces

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
T. Brocker, T. Dieck, Representations of compact Lie groups
A. Knapp, Lie groups beyond an introduction

Syllabus

2/25: Outline of the course, Symplectic vector space
2/26: symplectic linear reduction (除非特別說，不然時間都是10:30-11:20)
3/4: Compatible complex structure, linear symplectomorphism
3/5: Symplectic manifolds
3/11: Symplectic manifolds
3/12: cotangent bundles (assuming knowledge of differential forms and vector fields on manifolds)
3/18: group actions on manifolds, Hamiltonian actions
3/19: Hamiltonian actions
3/25: Hamiltonian actions and examples
3/26: More examples
4/1: 2d gauge theory as an example
4/2: University Holiday
4/8: Fibration, submersion
4/9: Foliation and distribution
4/15: 時間移到5/28 以及 6/4
4/16: 時間移到6/11

4/22: symplectic reduction of constant rank submanifolds
4/23: Symplectic reduction
4/29: Midterm
4/30: 時間移到6/11
5/6: Symplectic reduction
5/7: Reduction in stages
5/13: Reduction in stages
5/14: Morse theory and Convexity theory
5/20: Morse theory and Convexity theory
5/21: Catch up
5/27: Delzant polytope
5/28: Delzant theorem (時間注意是34堂)
6/3: Delzant theorem
6/4: Delzant theorem(時間注意是34堂)
6/10: 期末報告
6/11: 補課/期末報告 (時間注意是34堂)

Evaluation

Midterm 40%, Final presentation 60% (報告希望可以控制在50分鐘左右)

Last Updated: Feb 23, 2026
URL: http://www.math.nthu.edu.tw/~nankuo/ISG2026S.html