| Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
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Course: Special Topics - Introduction to Symplectic Geometry Instructor: Soumyashant Nayak Room: Physics Lab Level: Postgraduate Time: Currently offered |
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| Syllabus Past Exams Syllabus: Pre-Requisites :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 (i) An Introduction to Symplectic Geometry by R. Berndt (BER). (ii) Mathematical Methods in Classical Mechanics by V. I. Arnold (ARN). (iii) Geometry, Topology and Physics by M. Nakahara (NAK). Secondary References (i) Foundations of Mechanics by R. Abraham and J. Marsden (AM). (ii) 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). Evaluation:
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[ Semester Schedule ][ SMU ] [Indian Statistical Institute] |