Introduction to Symplectic Geometry (2023 Spring)

Room: 綜三 631
Time: Friday n56 (12:30-3:20)
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. 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

2/17: Symplectic vector space, symplectic linear reduction
2/24: University Holiday (also 理學院會議)
3/3: Compatible complex structure, linear symplectomorphism, Symplectic manifolds
3/10: Symplectic manifolds, cotangent bundles (assuming knowledge of differential forms and vector fields on manifolds)
3/17: group actions on manifolds, Hamiltonian actions
3/24: Hamiltonian actions and examples
3/31: 2d gauge theory as an example
4/7: Fibration, submersion, foliation and distribution, and the symplectic reduction of constant rank submanifolds

4/14: Symplectic reduction
4/21: Midterm
4/28: Reduction in stages
5/5: Morse theory and Convexity theory
5/12: Convexity theory and Delzant polytope
5/19: Delzant theorem notes
5/26: 報告(Fedosov's Quantization on Symplectic manifold) and moment maps
6/2: 報告 (Floer homology)
6/9*: 報告 and Darboux theorem

Evaluation

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

Last Updated: June 2, 2023
URL: http://www.math.nthu.edu.tw/~nankuo/ISG2023S.html