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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 2012 (Monday, October 8 to Friday, December 14)

Timetable

  • Tue 11:05 - 11:55

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
Photo of Dirk Schuetz


Students


Photo of Nada Alhabib
Nada Alhabib
(Liverpool)
Photo of Reem Alomair
Reem Alomair
(Manchester)
Photo of Steven Charlton
Steven Charlton
(Durham)
Photo of Frederico De Oliveira
Frederico De Oliveira
(East Anglia)
Photo of Chris Draper
Chris Draper
(York)
Photo of Michela Egidi
Michela Egidi
(Durham)
Photo of Jonathan Grant
Jonathan Grant
(Durham)
Photo of Serena Liu
Serena Liu
(Durham)
Photo of Callan McGill
Callan McGill
(Sheffield)
Photo of Daniel Rust
Daniel Rust
(Leicester)
Photo of Yafet Sanchez Sanchez
Yafet Sanchez Sanchez
(Southampton)
Photo of Thomas Sutton
Thomas Sutton
(Sheffield)
Photo of James Walton
James Walton
(Leicester)


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


Note:

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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 between 7th and 18th of January 2013. The examination will consists of five questions, and you will need to obtain 40% to pass the course.

Final Examination for Differential Topology and Morse Theory

Files:Exam paper
Deadline: Friday 18 January 2013 (1952.9 days ago)
Instructions:

There will be a take-home examination for this course. The file will be made available on 7 January 2013. The exam consists of five questions. All five 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 18 January.



Recorded Lectures


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