課程大綱

課程資訊

課程名稱

辛幾何導論
Introduction to Symplectic Geometry

開課學期

102-1

授課對象

理學院數學系

授課教師

蔡忠潤

課號

MATH5346

課程識別碼

221 U6190

班次

學分

全/半年

半年

必/選修

選修

上課時間

星期一3(10:20~11:10)星期四7,8(14:20~16:20)

上課地點

天數204天數204

備註

總人數上限：20人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1021sg

課程簡介影片

核心能力關聯

本課程尚未建立核心能力關連

課程大綱

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課程概述

1. Linear symplectic algebra.
2. Symplectic manifolds.
3. Normal form theorems.
4. Symplectic group actions, moment maps and symplectic reductions.
5. Atiyah--Guillemin--Sternberg convexity theorem.
6. Symplectic capacities.
7. Constructions of symplectic manifolds.
8. Some contact geometry.

課程目標

Symplectic geometry is one of the main branch in geometry over the past 20 years. This course
aims to give an introduction on symplectic manifold.

課程要求

1. General Topology (topological spaces, product topology, quotient topology and quotient maps,
continuity, compactness, connectedness).
2. Differentiable manifolds (tangent spaces, differential maps, differential forms).

預期每週課後學習時數

Office Hours

每週五 16:00~17:00

指定閱讀

參考書目

1. Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology.
2. Ana Cannas da Silva, Lectures on symplectic geometry.
3. Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics.
4. Hansjorg Geiges, An introduction to contact topology.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Homework	30%
2.	Midterm	30%
3.	Final exam/report	40%

課程進度

週次	日期	單元主題
第2週	9/16	symplectic manifold; tautological 1-form and canonical symplectic form on the cotangent bundle. Reference: [CdS1, §2]
第5週	10/07	Moser's trick, Darboux coordinate. Reference: [CdS1, §7.3, §8.1, §8.2].
第1-1週	9/09	origin of symplectic geometry, from Lagrangian mechanics to Hamiltonian mechanics. Reference: [M&S, p.12~15]
第1-2週	9/12	symplectic linear algebra, symplectic group, Lagrangian subspaces. Reference: [CdS1, §1], see also [M&S, p.19~21 and p.50~51]
第3-1週	9/23	Lagrangian submanifolds in cotangent bundle. Reference: [CdS1, §3]
第3-2週	9/26	construct symplectomorphism between cotangent bundles by generating functions. billiards. critical points of generating function and fixed points of symplectomorphism. Reference: [CdS1, §4, §5]
第4-1週	9/30	the Poincare--Birkhoff theorem. Reference: [M&S, §8.2]
第4-2週	10/03	isotopy, Cartan formula, chain homotopy induced by isotopy, Moser's trick. Reference: [CdS1, §6, §7]
第6-1週	10/14	Weinstein tubular neighborhood theorem. Reference: [CdS1, §8.3, §9].
第6-2週	10/17	Lie algebra of the group of symplectomorphisms, Hamiltonian and symplectic vector fields, fixed points of Hamiltonian flows, Poisson bracket, integrable systems. Reference: [CdS1, §9.3, §18].
第7-1週	10/21	integrable systems, Legendre transform. Reference: [CdS1, §18.4, §20.1].
第7-2週	10/24	Legendre transform , action-angle coordinate, group action. Reference: [CdS1, §20, §21].
第8-1週	10/28	a little Lie theory, moment map. Reference: [CdS1, §21.5, §22.1].
第8-2週	10/31	moment map, symplectic reduction. Reference: [CdS1, §22, §23].
第9-1週	11/04	more on the reduction, reduction in stages. Reference: [CdS1, §24].
第9-2週	10/07	MIDTERM
第10-1週	11/11	Lie algebra cohomology, existence of moment maps. [CdS1, §26.2, §26.3]
第10-2週	11/14	uniqueness of moment maps, compatible triple, convexity: linear convexity. Reference:[CdS1, §26.4, §12.2] and [GS, §32].
第11-1週	11/18	properties of group actions. Reference:[G&S, §27].
第11-2週	11/21	convexity: local convexity, global convexity, Morse--Bott. Reference:[G&S, §32].
第12-1週	11/25	more on the convexity, symplectic toric manifold. Reference:[G&S, §32] and [CdS1, §27].
第12-2週	11/28	cancelled
第13-1週	12/02	cancelled due to the Hsu Lectures in Academia Sinica
第13-2週	12/05	Delzant polytope, construction of toric manifolds from a Delzant polytope. Reference:[CdS1, §28].
第14-1週	12/09	Delzant polytope, examples of dimension two with four vertices, Hirzebruch surfaces. Reference:[CdS1, §28].
第14-2週	12/12	almost complex structure, compatible triple. Reference:[CdS1, §12 ~ §14].
第15-1週	12/16	topological properties of Kahler manifolds. Reference:[CdS1, §17].
第15-2週	12/19	cancelled
第16-1週	12/23	fibered three manifold, Kodaira--Thurston manifold. Reference:[CdS1, §17].
第16-2週	12/26	construction, fiber sum for symplectic manifolds. Reference: the article of Gompf, see also [M&S, §7.2].
第17-1週	12/30	the elliptic surface E(1). Reference: the article of Gompf, see also [M&S, Example 7.6].
第17-2週	1/02	symplectic four manifold with any finitely presentable group as its fundamental group. Reference: the article of Gompf, see also [M&S, §7.2].