Lectures on Symplectic Geometry

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Lectures on Symplectic Geometry
Ana Cannas da Silva


Table of Contents 

I  Symplectic Manifolds

1 Symplectic Forms

2 Symplectic Form on the Cotangent Bundle

II  Symplectomorphisms

3 Lagrangian Submanifolds

4 Generating Functions

5 Recurrence

III  Local Forms

6 Preparation for the Local Theory

7 Moser Theorems

8 Darboux-Moser-Weinstein Theory

9 Weinstein Tubular Neighborhood Theorem

IV  Contact Manifolds

10 Contact Forms

11 Contact Dynamics

V Compatible Almost Complex Structures

12 Almost Complex Structures

13 Compatible Triples

14 Dolbeault Theory

VI  Kahler Manifolds

15 Complex Manifolds

16 Kahler Forms

17 Compact Kahler Manifolds

VII  Hamiltonian Mechanics

18 Hamiltonian Vector Fields

19 Variational Principles

20 Legendre Transform

VIII  Moment Maps

21 Actions

22 Hamiltonian Actions

IX  Symplectic Reduction

23 The Marsden-Weinstein-Meyer Theorem

24 Reduction

X  Moment Maps Revisited

25 Moment Map in Gauge Theory

26 Existence and Uniqueness of Moment Maps

27 Convexity

XI  Symplectic Toric Manifolds

28 Classification of Symplectic Toric Manifolds

29 Delzant Construction

30 Duistermaat-Heckman Theorems



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