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Regular version of the site
2020/2021

Research Seminar "Topology I"

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type: Optional course (faculty)
When: 1, 2 module
Open to: students of all HSE University campuses
Language: English
ECTS credits: 6

Course Syllabus

Abstract

Topology studies a fairly general notion of space equipped with a notion of closeness. It is of importance in geometry and analysis. The course covers basic point-set topology, the classification of surfaces, and the theory of fundamental groups and covering spaces.
Learning Objectives

Learning Objectives

  • The lectures cover basis material and examples, while the seminars allow students to more deeply explore the theory and examples.
Expected Learning Outcomes

Expected Learning Outcomes

  • Students will become fluent in the basic notions of topology and will gain experience in the oral and written presentation of mathematics.
Course Contents

Course Contents

  • Basic point-set topology
    Definition of topological space, open sets, closed sets, continuous functions
  • Properties of topological spaces
    Hausdorff, connected, compact
  • Constructions of topological spaces
    Quotients, products, pushouts, pullbacks
  • Fundamental groups and covering spaces
    Definition of the fundamental group of a space. Relation between subgroups of the fundamental group and covering spaces. Calculations of fundamental groups using the van Kampen theorem.
  • Classification of surfaces
    Classification of compact 2-dimensional manifolds with boundary.
Assessment Elements

Assessment Elements

  • non-blocking Midterm exam
  • non-blocking Final Exam
  • non-blocking Seminar participation
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.4 * Final Exam + 0.4 * Midterm exam + 0.2 * Seminar participation
Bibliography

Bibliography

Recommended Core Bibliography

  • Наглядная топология, Прасолов, В. В., 2006

Recommended Additional Bibliography

  • Brown, R. (2006). Chapter 9: Computation of the fundamental groupoid: 9.1 The Van Kampen theorem for adjunction spaces. In Topology & Groupoids (pp. 339–352). Ronald Brown.