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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

• 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

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

#### 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

• Final exam
• Test
• Homework

#### Промежуточная аттестация

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