Michael Farber: Morse theory, Topology & Robotics

Lecture 1-Topology of sublevel sets, operation of attaching cells, Morse functions,

Morse Lemma, existence of Morse functions, crossing a Morse critical value. Main

theorem of Morse theory. Morse inequalities. Examples: spheres, complex projective

spaces, etc.

Lecture 2- Smale’s solution of Generalised Poincare Conjecture.

H-cobordism theorem. Ideas and techniques of the proof.

Lecture 3- Planar linkages and their configuration spaces. Formula for their Betti

numbers. Walker’s Conjecture.

Lecture 4- Spaces of polygons in high dimensions, classification of these spaces in

combinatorial terms.

Lecture 5- Robot motion planning and topology. Schwarz genus of a fibration. Lusternik-

Schnirelmann theory. Topological complexity.

- Recommended reading:

1. J. Milnor, Morse theory. 1963

2. J. Milnor, Lectures on the h-cobordism theorem. 1965.

3. M. Farber, Invitation to topological robotics, EMS, 2008.

- Additional Optional reading:

M. Farber and V. Fromm, The topology of spaces of polygons. Trans. Amer. Math. Soc.

365 (2013), no. 6, 3097–3114.

- Prerequisites:

Basic homology theory of cell complexes, basic knowledge smooth manifolds.

Lecturer Details:

Lecturer: Professor Michael Farber

Lecturer home institution: Queen Mary University of London

Lecturer e-mail: M.Farber@qmul.ac.uk

Lecturer telephone number: 07906345551