MAGIC002: Differential topology and Morse theory

Course details

A specialist MAGIC course

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 (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. 

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

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

The assessment for this course will be released on Monday 10th May 2021 at 00:00 and is due in before Monday 24th May 2021 at 11:00.

Assessment is via take-home examination. 

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

Files

Only current consortium members and subscribers have access to these files.

Please log in to view course materials.

Lectures

Please log in to view lecture recordings.