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MA6211 - Topics in Mathematics (Morse Theory)

Semester 1, 2013/2014

Every mathematician has a secret weapon. Mine is Morse theory. - Raoul Bott

Mathematics is an art, and as an art chooses beauty and freedom. It is an aid to technology, but is not a part of technology. It is a handmaiden of the arts, but it is not for this reason an art. Mathematics is an art because its mode of discovery and its inner life are like those of the arts. - Marston Morse

Course overview

Originally developed by Morse in the 1920s, the ideas from Morse theory have since been generalised and applied to many different areas of mathematics; for example Smale's proof of the Poincare conjecture in higher dimensions and Bott's work on Lie groups.

This module is meant for graduate students in Mathematics who are familiar with Geometry and Topology. The first part of the course will be an introduction to Morse theory and an overview of classical results due to Morse, Bott and Smale. The second part of the course will focus on more recent ideas in Morse theory due to Witten, Austin-Braam and Harvey-Lawson.

Lecture Schedule

Fridays 7pm-10pm in S17 05-12.
Click here for more detailed information about the lectures.

Aims and objectives

  1. (First half of the course) To explain the ideas of classical Morse theory (due to Morse) and related results by Bott and Smale.
  2. (Second half of the course) To explain modern ideas of Morse theory developed by Witten, Austin-Braam and Harvey-Lawson.

Objectives. By the end of the course, everyone should be able to:
  1. Use Morse theory and Morse-Bott theory to compute the homology of some basic examples
  2. Understand the Morse inequalities and the different criteria for a Morse function to be perfect
  3. Write down the Morse complex and compute the cohomology ring for some basic examples
  4. Appreciate the different ways that Morse theory has been used to prove important theorems


The grade will be based on 100% continuous assessment.
60% of the continuous assessment will be based on problems given throughout the semester. Half of this will be based on written solutions and half will be based on student presentation of solutions. The problems will be assigned regularly and the written solutions will be due in two parts: one on Friday 4 October (for problems based on the first half of the course) and one on Friday 8 November (for problems based on the second half of the course).
The remaining 40% of the continuous assessment will be based on student presentations on a topic of their choice at the end of the semester.

Text and readings

During the first half of the course we will cover Part I of "Morse theory" by John Milnor in detail and a selection of topics from the papers "Nondegenerate critical manifolds" by Raoul Bott and "Generalized Poincare's conjecture in dimensions greater than four" by Stephen Smale.
During the second half of the course we will cover a selection of topics from the papers "Supersymmetry and Morse theory" by Witten, "Morse-Bott theory and equivariant cohomology" by Austin and Braam and "Finite volume flows and Morse theory" by Harvey and Lawson.

Last updated 6 August, 2013