2018/2019
Научно-исследовательский семинар "Введение в симплектическую и контактную геометрию"
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Язык:
английский
Кредиты:
5
Контактные часы:
76
Course Syllabus
Abstract
The course is centered on the notion of symplectic structure, contact structure, legendrian and contact manifolds. In each direction we start from the very basic definition. Course is optional. Pre-requisites: Introductory courses on differential manifolds and ordinary differential equations.
Expected Learning Outcomes
- Can master mathematical methods that provide a broad view towards problems of classical mechanics, and apply these methods to other areas of pure mathematics and mathematical physics
Course Contents
- Linear symplectic geometry
- Symplectic structure, Hamiltonian fields, Darboux theorem
- Symplectic reduction
- Lagrangian manifolds, Maslov index
- First steps of symplectic topologyMorse theory, Arnold's conjectures, idea of Floer Homology
- Extra TheoremsLiouville - Arnold theorem, moment map. The Atiyah - Guillemin - Sternberg convexity theorem
- Distributions, integrability, contact structures. Examples. One jets of functions
- Legendrian manifolds
- Contact geometry and first-order partial differential equations
- Differential geometry of plane curves and contact structure
- Generating families
- An introduction to Lagrangian and Legendrian singularities
Bibliography
Recommended Core Bibliography
- Friedman, A. (2007). Advanced Calculus (Vol. Dover edition). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1153250
Recommended Additional Bibliography
- McLean, M. (2010). A spectral sequence for symplectic homology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1011.2478