General
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.
Semester
Autumn 2011 (Monday, October 10 to Friday, December 16)
Timetable- Tue 11:05 - 11:55
Students
|
Robin Allan
(Sheffield) |
Suliman Alsaeed
(Liverpool) |
Robert Friswell
(York) |
Daniel Jones
(Durham) |
|
Benjamin Lang
(York) |
Vladimir Lukiyanov
(Loughborough) |
Liangang Ma
(Liverpool) |
Vasiliki Nassiopoulou
(Leeds) |
|
Rodrigo Pires dos Santos
(Leeds) |
Simon StJohn-Green
(Southampton) |
Dragomir Tsonev
(Loughborough) |
Kenneth Uda
(Loughborough) |
|
Marco Wong
(Leeds) |
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.
Bibliography
| Lectures on the h-cobordism theorem | Milnor |
| An Invitation to Morse Theory | Nicolaescu |
| Morse theory | Milnor |
| Lectures on Morse homology | Banyaga and Hurtubise |
| Topology and Geometry | Bredon |
| Foundations of differentiable manifolds and Lie groups | Warner |
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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 via a take-home examination, which will be made available shortly after the end of the course on 13 December. The examination will consists of five questions, and you will need to obtain 40% to pass the course. The deadline for completing the examination is 13 January 2012.
Assignments
Final Examination
| Files: | Exam paper | ||
| Deadline: | Friday 13 January 2012 (497.2 days ago) | ||
| Instructions: | See file for instructions. | ||
Files
Files marked L are intended to be displayed on the main screen during lectures.