Fundamentals of algebra and differential geometry

Everyone is welcome, including interested undergraduates.

Read at 8-st semester.
2 hours of lectures per week

Analysis on smooth manifolds plays an important role in modern mathematics and theoretical physics in the study of objects that do not have a natural linear structure. An example of such objects is a Lie group, which first appeared in the late 19th century in the works of the Norwegian mathematician Sophus Lie symmetries in the analysis of differential equations and developed, since an independent mathematical discipline needed in the study of dynamical systems with explicit or implicit symmetry. Dynamical systems on smooth manifolds often arise when the symmetry reduction of dynamical systems on linear spaces.

Study analysis on smooth manifolds also promotes the formation of mathematical intuition, not related to a specific coordinate system, and allows you to look in more general terms on the facts learned in the general course of mathematical analysis.

To understand the course requires a good knowledge of general math course of the first four semesters, as well as an introduction of general topology. Due to lack of time, not part of the allegations proved. The students are offered tasks that need to get credit.

• Shchepetilov Alexei Valeriyovych

1. Algebra external forms in linear space. Hodge operator in the algebra of exterior forms on the pseudo-Euclidean space.
2. Smooth and topological manifolds. Classification of two-dimensional compact topological manifolds.
3. Vector bundles. Parallelizable manifold. Neparallelizuemost even-dimensional spheres.
4. Exterior differential. Lie derivative. Maxwell equations in the language of exterior forms.
5. Integration of differential forms. Stokes formula.
6. Basics of Hamiltonian mechanics on symplectic manifolds.
7. Lie groups and Lie algebras.
8. Actions of Lie groups on smooth manifolds.
9. Universal enveloping algebra of the Lie algebra.
10. Poisson structure on the symmetric algebra S (g) of the Lie algebra g.
11. Poisson action of a Lie group and the moment map.
12. The method of Hamiltonian reduction of Hamiltonian systems with symmetry.
13. Linear connections on vector bundles.
14. Covariant derivative and the curvature of the linear connection on a vector bundle.

Литература.

• Основная.
  1. Арнольд В.И. Математические методы классической механики. М.Наука. 1974, 1979, 1989.
  2. Спивак М. Математический анализ на многообразиях. М.: Мир. 1968.
  3. Уорнер Ф. Основы теории гладких многообразий и групп Ли. М. Мир.1987.
  4. Винберг Е.Б. Курс алгебры. М. Факториал Пресс. 2002.
• Дополнительная.
  1. Шафаревич И.Р. Основные понятия алгебры. // В книге: Итоги науки и техники. Серия: Современные проблемы математики. Фундаментальные направления. Т. 11. М. ВИНИТИ. 1986; РХД, Москва-Ижевск, 2000.
  2. Дубровин Б.А., Новиков С.П., Фоменко А.Т. Современная геометрия: методы и приложения. М. Наука. 1979, 1986.
  3. Постников М.М. Гладкие многообразия. М. Наука. 1987.
  4. Постников М.М. Дифференциальная геометрия. М. Наука. 1988.
  5. Постников М.М. Группы и алгебры Ли. М. Наука. 1982.
  6. Шапуков Б.Н. Задачи по группам Ли и их приложениям. М. РХД. 2002.

Questions to settle on a special course