Differential Topology

10 credits

Syllabus, Master's level, 1MA040

A revised version of the syllabus is available.
Code
1MA040
Education cycle
Second cycle
Main field(s) of study and in-depth level
Mathematics A1F
Grading system
Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
Finalised by
The Faculty Board of Science and Technology, 15 March 2007
Responsible department
Department of Mathematics
What is a course syllabus?

Entry requirements

BSc, Topology

Learning outcomes

In order to pass the course (grade 3) the student should be able to

  • give an account of important concepts and definitions in the area of the course;

  • exemplify and interpret important concepts in specific cases;

  • formulate important results and theorems covered by the course;

  • describe the main features of the proofs of important theorems;

  • express problems from relevant areas of applications in a mathematical form suitable for further analysis;

  • use the theory, methods and techniques of the course to solve mathematical problems;

  • present mathematical arguments to others.

    Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.

    Content

    Manifolds, submanifolds, smooth maps, tangent space, tangent bundles. The constant rank theorem. Immersions and embeddings. Submersions. Transversality. Sard's theorem. Brouwer's fix point theorem. Intersection theory modulo 2. Jordan–Brouwer's separation theorem. Orientation. Oriented intersection theory. Lefschetz's fix point theorem. The degree of a map. Hopf's degree theorem. Vector fields on manifolds and Poincaré–Hopf's theorem. Morse functions. Classification of one- and two-dimensional manifolds.

    Instruction

    Lectures and problem solving sessions.

    Assessment

    Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.