MAGIC002: Differential topology and Morse theory

Course details

Semester

Autumn 2017
Monday, October 9th to Friday, December 15th

Hours

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

Timetable

Wednesdays
13:05 - 13:55

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

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Assessment

The assessment for this course will be released on Monday 8th January 2018 and is due in by Sunday 21st January 2018 at 23:59.

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 and a half questions to pass the course.

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

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Lectures

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