Бакалавриат
2025/2026
Проектный семинар "Дифференциальная геометрия и топология"
ID 1095169
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Курс обязательный (Физика)
Кто читает:
Департамент физики
Где читается:
Школа информатики, физики и технологий
Когда читается:
3-й курс, 3, 4 модуль
Охват аудитории:
для своего кампуса
Преподаватели:
Постушков Сергей Сергеевич
Язык:
английский
Кредиты:
5
Контактные часы:
72
Course Syllabus
Abstract
The project seminar is aimed at studying issues related to the qualitative properties and differences of objects considered in geometry, analysis, mathematical physics, theory of differential equations and other mathematical and physical disciplines. To master the discipline, the basics of mathematical analysis of many variables, algebra and group theory are required.
Learning Objectives
- To introduce students to topology and differential geometry, and to help them master the specific nature of these mathematical fields and their research methodologies.
- To develop practical skills in solving problems that arise within these disciplines.
- To apply topology and differential geometry as a core framework for constructing models in physics.
Expected Learning Outcomes
- Demonstrates the ability to solve problems arising within the fields of topology and differential geometry.
- Utilizes the language of fiber bundles and connections as a framework for building models in physics.
- Exhibits the ability to apply category theory language to mathematical problems.
- Formulates basic definitions and key results in topology and differential geometry.
- Demonstrates the ability to manipulate the fundamental objects and constructions of topology and differential geometry.
- Utilizes various tools to compute the fundamental group of a topological space, including its functoriality, results from covering space theory, and the Seifert–van Kampen theorem.
- Demonstrates core skills in planning, structuring, and delivering an effective talk.
Course Contents
- General topology
- Category theory
- Homotopy theory (fundamental group)
- Smooth manifolds
- Fiber bundles and connections on them
Assessment Elements
- Seminar presentation 1
- ColloquiumColloquium at the end of 3d module.
- Seminar presentation 2
- Exam
Interim Assessment
- 2025/2026 4th module0.25 * Exam + 0.25 * Seminar presentation 2 + 0.25 * Colloquium + 0.25 * Seminar presentation 1
Bibliography
Recommended Core Bibliography
- Algebraic Topology, 22nd printing, XII, 544 p., Hatcher, A., 2019
- Categories and sheaves, Kashiwara, M., 2006
- Differential and riemannian manifolds, Lang, S., 1995
- Topology, Munkres, J. R., 2000
- Дифференциальная геометрия и алгебры Ли и их приложения в теории поля, Волобуев, И. П., 2015
- Дифференциальные формы, расслоения, связности, Казарян, М. Э., 2002
Recommended Additional Bibliography
- Математика как метафора, Манин, Ю. И., 2008