Differential Topology

Vrije Universiteit Amsterdam

Course Description

  • Course Name

    Differential Topology

  • Host University

    Vrije Universiteit Amsterdam

  • Location

    Amsterdam, The Netherlands

  • Area of Study

    Algebra, Geometry, Mathematics

  • Language Level

    Taught In English

  • Course Level Recommendations

    Upper

    ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

    Hours & Credits

  • ECTS Credits

    6
  • Recommended U.S. Semester Credits
    3
  • Recommended U.S. Quarter Units
    4
  • Overview

    Course Objective
    At the end of this course:
    -the student understands the introduced concepts such as degree of a
    function, intersection number, coverings, and is able to use these to
    prove some fundamental results;
    -the student understands the meaning of the theorems and knows how to
    derive them;
    -the student can relate different properties to each other, for example
    relating the zeros of transversal vector fields to the genus of the
    surface;
    -the student knows examples of investigating the same mathematical
    content from different mathematical perspectives.


    Course Content
    Differential topology studies differentiable manifolds and
    differentiable functions from a topological viewpoint. In contrast to
    differential geometry properties arising from metrics are not studied,
    and in contrast to the general topology course, the spaces are
    restricted to nicely behaving manifolds. Studied properties of the
    manifolds are rather global than local. The classification of coverings
    can be seen as an alternative formulation of the fundamental theorem of
    Galois theory, allowing to derive an algebraic theorem from a
    topological perspective.

    The following topics will be covered during the course:
    -smooth manifolds;
    -classification of coverings;
    -Sard’s theorem;
    -Brouwer’s fixed point theorem;
    -transversality;
    -intersection theory and degree theory;
    -Euler characteristic;
    -index of vector fields;
    -Hopf theorem


    Teaching Methods
    Lectures and tutorials (2+2 hours per week)


    Method of Assessment
    For this course there is a midterm
    examination (50%) and a final exam (50%). There will also be a resit
    examination.


    Literature
    Guilluimin and Pollack: Differential Topology


    Target Audience
    Bachelor Mathematics, year 3


    Additional Information
    This course is offered for the first time. Master students are welcome
    to participate as well.


    Recommended background knowledge
    Topology
    Differential geometry

Course Disclaimer

Courses and course hours of instruction are subject to change.

Some courses may require additional fees.

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