Course page for Symplectic Geometry

This course is taught jointly in CMI and IMSc in Spring 2017 (Jan-April 2017).

Schedule : Monday, 3:30 pm at IMSc, Room 217     Friday, 2 pm at CMI, Lecture Hall 1.
Students registered for the class have to attend both lectures.

References: Lectures in Symplectic Geometry by Ana Cannas da Silva, Introduction to Symplectic topology by McDuff and Salamon.

What is happening in class?
(All chapter and section numbers are from Ana Cannas da Silva's book)

Lectures 1 and 2: Motivations of sympletic geometry from hamiltonian mechanics, statements of some important results of the subject. Chapters 1-3. Recommended reading: Section 1 and 2 of What is symplectic geometry by Dusa McDuff.
Attempt Homework 1.

Lectures 3 and 4: Moser and Darboux theorems. Chapters 6 and 7. We needed the statement of the Hodge theorem to prove Moser's theorem, so we covered Section 17.1. Homework 2 is due on Monday 23rd January.

Lecture 5 and 6: Weinstein's Lagrangian neighbourhood theorem and some applications. Section 8.3 and Chapter 9. Started discussing almost complex structures (Chapter 12). Nachiketa's presentation : contractability of the space of almost complex structures on a symplectic vector space. Homework 3 is due Friday 3rd February.

Lecture 7, 30th January: Lie group-theoretic proof of existence of compatible almost complex structure, and the contractibility of the space of almost complex structures. (Chapter 12 and 13)

Lecture 8, 3rd February: Integrability of almost complex structures : the statement of Newlander-Nirenberg theorem. Hamiltonian circle actions on symplectic manifolds -- some examples. Homework 4 is due Monday 13th February.

Lecture 9, 6th February: Motivating the moment map definition (Sections 18.1, 18.3, 26.1), Noether's theorem (section 24.1, 24.2), statement of symplectic reduction theorem (Marsden-Weinstein) section 23.1.

Lecture 10, 10th February: Recalled material from last lecture. Stated and proved Marsden-Weinstein theorem. Neetal's presentation: quotienting a manifold by a free action of a compact group gives a smooth manifold.

Lecture 11, 13th February: Defined Fubini-Studi form on projective space using symplectic reduction of Cn. Showed that the toric action on projective space is Hamiltonian.

Lecture 12, 17th February: Stated convexity theorem. More examples of moment maps : action of unitary group on complex vector spaces. Showed that complex grassmanians have a symplectic structure.

Homework 5 is due on Monday 27th February. Regular lectures were cancelled in mid-term week 20-24th Feb. We met once on Thursday 23rd February in CMI from 2-4:45pm.

Lecture 13, 23rd February: Ananyo showed that coadjoint orbits in the dual of the Lie algebra are symplectic manifolds with Hamiltonian group actions. Akashdeep described principal bundles, connections and curvature, and showed that for circle groups the gauge group action on connections is Hamiltonian.

Lecture 14, 27th February: Described Kähler forms as Hessian of plurisubharmonic function on complex manifolds -- section 16.3 in book. Homework 6 is due on Friday March 10th.

Lecture 15, 3rd March: Described the Fubini-study form on projective space using plurisubharmonic functions, and showed that this description agreed with that coming from symplectic reduction.

Lecture 16, 6th March: Symplectic forms on one-point blow-ups. See notes.

Lecture 17, 10th March: Repeated some material from previous lecture. Yash presented existence and uniqueness results for moment maps.

Lecture 18, 13th March: Symplectic cutting, and its effect on the moment polytope. Blowing down as symplectic sum. Introduced first Chern class.
Homework 7 is due on Monday March 20th.

Lecture 19, 17th March: First Chern class. Definition using curvature of a unitary connection. The first Chern number of a line bundle on a compact Riemann surface is the number of zeros of a section of the line bundle. Some more properties of the First Chern class.

Lecture 20, 20th March: Introduction to pseudoholomorphic curves on symplectic manifolds.
Homework 8 is due on Friday March 31st.

Lecture 21, 24th March: Pseudoholomorphic curves continued. Energy identity. Can the moduli space be compact? Statement of compactness theorem, when there is no 'smaller' spherical homology class.

Lecture 22, 27th March: Mohan's presentation on symplectic forms on symplectic fiber bundles.

Lecture 23, 31st March: Mohan finished presentation ending with construction of Kodaira-Thurston example of a symplectic manifold that is not Kahler. Sambit presented theorem on variation of reduced symplectic form as level set of an S¹ moment map is varied.

Lecture 24, 3rd April: Background on Morse-Bott functions. Paramjit presented proof of convexity theorem for image of moment maps.

Lecture 25, 7th April: Paramjit finished presentation. When does the set of J-holomorphic curves have a manifold structure?

Lecture 26, 10th April: Proof of Gromov's non-squeezing theorem.