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Introduction to Symplectic Geometry, Fall 2013

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Course Information

• Syllabus
• Information
• Ceiba
• Lectures: Monday, 10:20 ~ 11:10 and Thursday 15:30 ~ 17:20 at Astro-Math 204
• Office hours: Friday 16:00 ~ 17:00 or by appointment, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have three jokers: the lowest three grades will be discarded.
  • Midterm 30%
  • Final report 40%
• References:
  • [CdS1] Ana Cannas da Silva, Lectures on symplectic geometry. MR
    (I reserved this book in the library. The updated version can be downloaded from the author's website.)
  • [M&S] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology. MR
    (I reserved this book in the library. The errata can be found on the author's website.)
  • [CdS2] Ana Cannas da Silva, Symplectic toric manifolds. MR
    (This is a lecture note in a book of collected lectures. It can be found on the author's website.)
  • [G&S] Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics. MR
  • [G] Hansjörg Geiges, An introduction to contact topology. MR
    (I reserved this book in the library. This book is not the main reference of this course. The errata can be found on the author's website.)

Lecture summaries and references

• (9/9) origin of symplectic geometry, from Lagrangian mechanics to Hamiltonian mechanics. Reference: [M&S, p.12~15] and prelude.
• (9/12) symplectic linear algebra, symplectic group, Lagrangian subspaces. Reference: [CdS1, §1], [M&S, p.19~21 and p.50~51], see also note0912.
• (9/16) symplectic manifold; tautological 1-form and canonical symplectic form on the cotangent bundle. Reference: [CdS1, §2].
• (9/23) Lagrangian submanifold in cotangent bundle. Reference: [CdS1, §3].
• (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].
• (9/30) the Poincaré--Birkhoff theorem. Reference: [M&S, §8.2], see also note0930.
• (10/3) isotopy, Cartan formula, chain homotopy induced by isotopy, Moser's trick. Reference: [CdS1, §6, §7].
• (10/7) Moser's trick, Darboux coordinate. Reference: [CdS1, §7.3, §8.1, §8.2].
• (10/14) Weinstein tubular neighborhood theorem. Reference: [CdS1, §8.3, §9].
• (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].
• (10/21) integrable systems, Legendre transform. Reference: [CdS1, §18.4, §20.1].
• (10/24) Legendre transform, action-angle coordinate, group action. Reference: [CdS1, §20, §21].
• (10/28) a little Lie theory, moment map. Reference: [CdS1, §21.5, §22.1].
• (10/31) moment map, symplectic reduction. Reference: [CdS1, §22, §23].
• (11/4) more on the reduction, reduction in stages. Reference: [CdS1, §24].
• (11/7) Midterm: solution.
• (11/11) Lie algebra cohomology, existence of moment maps. Reference: [CdS1, §26.2, §26.3].
• (11/14) uniqueness of moment maps, compatible triple, convexity: linear convexity. Reference: [CdS1, §26.2, §12.3] and [G&S, §32].
• (11/18) properties of group actions. Reference:[G&S, §27].
• (11/21) convexity: local convexity, global convexity, Morse--Bott. Reference:[G&S, §32].
• (11/25) more on the convexity, symplectic toric manifold. Reference:[G&S, §32] and [CdS1, §27].
• (12/2) class cancelled due to the Hsu Lectures.
• (12/5) Delzant polytope, construction of toric manifolds from a Delzant polytope. Reference:[CdS1, §28], see also lecture1205.
• (12/9) Delzant polytope, examples of dimension two with four vertices, Hirzebruch surfaces. Reference:[CdS1, §28].
• (12/12) almost complex structure, compatible triple. Reference:[CdS1, §12 ~ §14].
• (12/16) topological properties of Kähler manifolds. Reference:[CdS1, §17].
• (12/23) fibered three manifold, Kodaira--Thurston manifold. Reference:[CdS1, §17].
• (12/26) more on the fibered three manifold construction, fiber sum for symplectic manifolds. Reference: the article of Gompf, see also [M&S, §7.2].
• (12/30) the elliptic surface E(1). Reference: the article of Gompf, see also [M&S, Example 7.6].
• (1/2) symplectic four manifold with any finitely presentable group as its fundamental group. Reference: the article of Gompf, see also [M&S, §7.2].

Homework

• Homework 01: due Monday, September 16.
• Homework 02: due Monday, September 23.
• Homework 03: due Monday, September 30.
• Homework 04: due Monday, October 7.
• Homework 05: due Monday, October 14.
• Homework 06: due Monday, October 21.
• Homework 07: due Monday, October 28.
• Homework 08: due Monday, November 4. See also note1104.
• Homework 09: due Monday, November 18.
• Homework 10: due Monday, November 25.
• Homework 11: due Monday, December 2. The deadline is rescheduled to December 5.
• Homework 12: due Monday, December 9.
• Homework 13: due Monday, December 16.
• Homework 14: due Monday, December 23.

Final report

• The oral presentation will take place in Astro-Math 305. The written report is due 17:00, January 17.
• You are required to attend the final report of three others.
• Office hour of the last week: 14:00 ~ 15:00 on Friday, January 3; 19:30 ~ 20:30 on Wednesday, January 8; 16:10 ~ 17:10 on Friday, January 10.

January	13	14	15
9:30	Chao	Hsu	Chiu
10:30	Ooi	Kuo	Payne
14:00	Chen	H.-P. Wang	C.-H. Wang
15:00