MAGIC002: Differential topology and Morse theory

Course details

A specialist MAGIC course

Semester

Spring 2020
Monday, January 20th to Friday, March 27th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Mondays
09:05 - 09:55 (UK)

Description

The course will give an introduction to Morse Theory. This theory studies the topology of smooth manifolds through real-valued smooth functions whose critical points satisfy a certain non-degeneracy condition. We will investigate how the homotopy type is related to critical points and how the homology of a manifold can be calculated through Morse functions.

Prerequisites

Basic knowledge of Differentiable Manifolds and Algebraic Topology is necessary. This can be obtained through the Core Courses MAGIC063 and MAGIC064.

Related courses

Syllabus

  • Smooth functions, non-degenerate critical points, Morse functions.
  • Morse Lemma.
  • Morse functions on spheres, projective spaces, orthogonal groups, configuration spaces of linkages.
  • Homotopy type, cell decompositions of manifolds.
  • Existence of Morse functions, cobordisms.
  • gradient flows, stable and unstable manifolds.
  • resonant Morse functions, ordered Morse functions.
  • Morse homology, Morse inequalities.
  • Calculations for projective spaces.
  • Introduction to the h-cobordism theorem.

Lecturer

  • DS

    Dr Dirk Schuetz

    University
    Durham University

Bibliography

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Assessment

The assessment for this course will be released on Monday 20th April 2020 at 00:00 and is due in before Monday 4th May 2020 at 11:00.

Assessment is via take-home examination. There will be four questions. The best three answers will count towards pass/fail, and you will need 50% to pass the examination.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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