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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 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Wed 13:05 - 13:55

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


Dirk Schuetz
Email dirk.schuetz@durham.ac.uk
Phone (0191) 334 3089
vcard
Photo of Dirk Schuetz


Students


Photo of Hussien Abugirda
Hussien Abugirda
(Reading)
Photo of Birzhan Ayanbayev
Birzhan Ayanbayev
(Reading)
Photo of John Blackman
John Blackman
(Durham)
Photo of Josh Cork
Josh Cork
(Leeds)
Photo of Lorenzo De Biase
Lorenzo De Biase
(Cardiff)
Photo of Joe Driscoll
Joe Driscoll
(Leeds)
Photo of Dominic Foord
Dominic Foord
(Liverpool)
Photo of Massimo Gisonni
Massimo Gisonni
(Loughborough)
Photo of Megan Goode
Megan Goode
(*External)
Photo of Ai Guan
Ai Guan
(Lancaster)
Photo of Thomas Honey
Thomas Honey
(Manchester)
Photo of Sarah Liddell
Sarah Liddell
(Leeds)
Photo of Robert Little
Robert Little
(Durham)
Photo of Joe Oliver
Joe Oliver
(Leeds)
Photo of Edward Pearce
Edward Pearce
(Sheffield)
Photo of Luca Pol
Luca Pol
(Sheffield)
Photo of Gregory Roberts
Gregory Roberts
(Liverpool)
Photo of David Robertson
David Robertson
(Newcastle)
Photo of Raul Sanchez Galan
Raul Sanchez Galan
(*External)
Photo of Motiejus Valiunas
Motiejus Valiunas
(Southampton)
Photo of Jordan Williamson
Jordan Williamson
(Sheffield)
Photo of Albert Wood
Albert Wood
(*External)


Bibliography


Lectures on the h-cobordism theoremMilnor
An Invitation to Morse TheoryNicolaescu
Morse theoryMilnor
Lectures on Morse homologyBanyaga and Hurtubise
Topology and GeometryBredon
Foundations of differentiable manifolds and Lie groupsWarner


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Assessment



The Assessment for this course will be via a take-home examination, to be taken during the official assessment period. The examination will consists of five questions, and you will need to obtain the equivalent of two questions to pass the course.

Final Exam of the Morse theory course

Files:Exam paper
Released: Monday 9 January 2017 (110.0 days ago)
Deadline: Sunday 22 January 2017 (96.0 days ago)
Instructions:

Answer all questions on separate pages. All fi ve questions carry the same mark. To pass you need to get 40% of the total score. You are allowed to use the materials provided through the course website. You need to upload the answers to the MAGIC website by 22 January.



Recorded Lectures


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