MAGIC002: Differential topology and Morse theory

Course details

A specialist MAGIC course

Semester

Autumn 2024
Monday, October 7th to Friday, December 13th

Hours

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

Timetable

Mondays
10:05 - 10:55 (UK)

Course forum

Visit the https://maths-magic.ac.uk/index.php/forums/magic002-differential-topology-and-morse-theory

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

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 13th January 2025 at 00:00 and is due in before Friday 24th January 2025 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

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.