MAGIC002: Differential topology and Morse theory

Course details

Semester

Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th

Hours

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

Timetable

Mondays
09:05 - 09:55

Course forum

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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. 

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

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Recorded Lectures

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