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Обычная версия сайта
2016/2017

Топология 2

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 5

Course Syllabus

Abstract

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
Learning Objectives

Learning Objectives

  • The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk
Expected Learning Outcomes

Expected Learning Outcomes

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

Course Contents

  • 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
Assessment Elements

Assessment Elements

  • non-blocking Final exam
  • non-blocking Test
  • non-blocking Homework
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.6 * Final exam + 0.2 * Homework + 0.2 * Test
Bibliography

Bibliography

Recommended Core Bibliography

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

Recommended Additional Bibliography

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