MAGIC002: Differential topology and Morse theory

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.

Basic knowledge of Differentiable Manifolds and Algebraic Topology is necessary.

This can be obtained through the Core Courses MAGIC063 and MAGIC064.

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.

MAGIC002: Differential topology and Morse theory

Course details

Semester

Hours

Timetable

Description

Prerequisites

Syllabus

Lecturer

Dr Dirk Schuetz

Bibliography

Assessment

Files

Lectures