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Обычная версия сайта
Версия для слабовидящихЛичный кабинет сотрудника ВШЭПоиск
2018/2019

Научно-исследовательский семинар "Введение в симплектическую и контактную геометрию"

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский

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.
Learning Objectives

Learning Objectives

  • To introduce the methods of symplectic and contact geometry
Expected Learning Outcomes

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

Course Contents

  • Linear symplectic geometry
  • Symplectic structure, Hamiltonian fields, Darboux theorem
  • Symplectic reduction
  • Lagrangian manifolds, Maslov index
  • First steps of symplectic topology
    Morse theory, Arnold's conjectures, idea of Floer Homology
  • Extra Theorems
    Liouville - 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
Assessment Elements

Assessment Elements

  • non-blocking Cumulative Grade
  • non-blocking Final Exam
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.3 * Cumulative Grade + 0.7 * Final Exam
Bibliography

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