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Обычная версия сайта
2017/2018

Топология 1

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

Программа дисциплины

Аннотация

Topology is a branch of mathematics that tries to answer the question whether two geometric shapes can be transformed into one another by stretching but without cutting or tearing. Quite surprisingly, these questions arise in almost every area of mathematics, from probability to algebra. So in particular, taking a graduate-level course in analysis or any kind of geometry requires some background in topology. On the other hand, topology in itself is a fascinating subject and it is full of surprises. This course is intended as an introduction to topology. We will first cover some point-set topology. We will focus on the material which we’ll need in the rest of the course and which is also likely to be a prerequisite for taking courses in other disciplines. Then we’ll look at the basics of algebraic and geometric topology and consider some applications
Цель освоения дисциплины

Цель освоения дисциплины

  • The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk
Результаты освоения дисциплины

Результаты освоения дисциплины

  • Successful participants improve their presentation skills and prepare for participation in research projects in the subject area
Содержание учебной дисциплины

Содержание учебной дисциплины

  • Basic definitions (topological spaces, continuous maps, compact spaces, separation axioms, quotient topology, homotopy between maps) and first applications (the main theorem of algebra)
  • Metric spaces. The completion of a metric space; Banach’s fixed point theorem. Compactness criteria for metric spaces. The Stone-Weierstrass theorem. The Hausdorff metric
  • The Euler characteristic and the classification of surfaces. Applications (the number of the ovals of a smooth real projective curve; regular polyhedra in 3-space, etc.) The Riemann-Hurwitz formula and some applications (the genus of a plane curve, the Hurwitz bounds on the order of the automorphism groups of complex curves). CW-complexes. Subcomplexes and quotient complexes. Homotopy extension property and applications
  • The fundamental group. Covering spaces and the correspondence between subgroups of the fundamental group and connected covering spaces of a given space
Элементы контроля

Элементы контроля

  • неблокирующий Created with Sketch. Final exam
  • неблокирующий Created with Sketch. Test
  • неблокирующий Created with Sketch. Homework
Промежуточная аттестация

Промежуточная аттестация

  • Промежуточная аттестация (4 модуль)
    0.6 * Final exam + 0.2 * Homework + 0.2 * Test
Список литературы

Список литературы

Рекомендуемая основная литература

  • Allen Hatcher. (2001). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.5FA4491E

Рекомендуемая дополнительная литература

  • James, I. M. Handbook of Algebraic Topology: North Holland: p.1324 , 1995. - ISBN 978-0-444-81779-2