Symplectic Geometry, Even Sem 2024-25
Introduction to Symplectic Geometry - Even Semester 2024-25
Term: Even Semester 2024-25
Duration: Jan 01 - Apr 30, 2025
Course Name: Introduction to Symplectic Geometry
Lecturer: Soumyashant Nayak
Lecture Schedule: Mon 2:00 p.m.-3:00 p.m.
Tue 10:00 a.m.-12:45 p.m. (with 15-30 min coffee break)
Room Allocation: TBD
Prerequisites: Basic familiarity with differentiable manifolds (Chapter 1 of "Foundations of Differentiable Manifolds and Lie Groups" by F. Warner), and matrix Lie groups (Chapters 1-5 of B. Hall's "Lie groups, Lie Algebras, and Representations: An elementary introduction".) These topics (and more) will be quickly reviewed in the initial phases of the course.
Primary References
- An Introduction to Symplectic Geometry by R. Berndt (BER).
- Mathematical Methods in Classical Mechanics by V. I. Arnold (ARN).
- Geometry, Topology and Physics by M. Nakahara (NAK).
Secondary References
- Foundations of Mechanics by R. Abraham and J. Marsden (AM).
- Symplectic Techniques in Physics by V. Guillemin and S. Sternberg.
Course Description
PHASE I (Preliminaries)
Analytical mechanics: Newtonian mechanics, Lagrangian formalism, Hamiltonian formalism; Canonical quantization; Path integral quantization of a harmonic oscillator.
Reference: NAK Chapter 1 (Quantum Physics), BER Chapter 0 (Some Aspects of Theoretical Mechanics)
Differentiable Manifolds: Heuristic introduction, Definition, Examples; Calculus on manifolds: Tangent spaces, cotangent spaces; Definition of vector bundle on manifolds, tangent and cotangent bundles; Tensor fields; Flows and Lie derivatives; Differential forms; Integration of differential forms; Lie groups and Lie algebras; Action of Lie groups on manifolds.
Reference: NAK Chapter 5 (Manifolds), ARN Chapter 7 (Differential Forms), BER Appendix A (Differentiable Manifolds and Vector Bundles)
Symplectic algebra: Symplectic vector spaces; symplectic morphisms and symplectic groups; subspaces of symplectic vector spaces; complex structures of real symplectic spaces.
Reference: BER Chapter 1 (Symplectic Algebra)
Estimated time: 24 hours of lectures (spread over 8 weeks)
PHASE II (Symplectic Manifolds and Momentum Mappings)
Symplectic manifolds: Definition and their morphisms; Hamiltonian vector fields and the Poisson bracket; Hamiltonian phase flows and their integral invariants; The Lie algebra of vector fields, The Lie algebra of hamiltonian functions; Symplectic coordinates and Darboux's theorem.
Reference: ARN Chapter 8 (Symplectic manifolds), BER Chapter 2 (Symplectic Manifolds), BER Chapter 3 (Hamiltonian vector fields and the Poisson bracket).
Complex manifolds: Definition, Calculus on complex manifolds; Complex differential forms; Hermitian manifolds and Hermitian differential geometry; Kähler manifolds and Kähler differential geometry.
Reference: NAK Chapter 8 (Complex Manifolds).
Examples of symplectic manifolds: The cotangent bundle, Kähler manifolds, coadjoint orbits, complex projective space.
Reference: BER Chapter 2 (Symplectic Manifolds).
The Momentum Mapping: Definition, Hamiltonian G-spaces; Constructions and examples; Noether's theorem.
Reference: BER Chapter 4 (The Moment Map), AM Chapter 4 Section 2 (The Momentum Mapping).
Estimated time: 18 hours (spread over 6 weeks)