# Applications of Algebraic Topology

Lead Research Organisation:
Durham University

Department Name: Mathematical Sciences

### Abstract

In this research we will study problems of algebraic topology which are inspired by applications. Consider a mechanical system which functions in the following regime: an operator introduces the current and the desired state of the system and the computer of the system (using an apriori designed motion planning algorithm) determines a continuous motion of the system from its current state to the desired state. In this research we study topological properties of such motion planning algorithms. In most cases such motion planning algorithms are discontinuous, i.e. small perturbations of the input data may lead to significant changes in the motion curve. The theory we plan to develop further in this reserach allows to use methods of algebraic topology to describe instabilities in the motion planning algorithms and to minimize them. We will also tackle several other interesting topological problems inspired by robotics applications.

### Organisations

## People |
## ORCID iD |

Michael Farber (Principal Investigator) |

### Publications

Barnett K
(2009)

*Topology of configuration space of two particles on a graph, I*in Algebraic & Geometric Topology
Cohen D
(2010)

*Topological complexity of collision-free motion planning on surfaces*in Compositio Mathematica
COSTA A
(2011)

*MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS*in Communications in Contemporary Mathematics
Farber M
(2010)

*Topology of configuration space of two particles on a graph, II*in Algebraic & Geometric Topology
Farber M
(2008)

*Robot motion planning, weights of cohomology classes, and cohomology operations*in Proceedings of the American Mathematical Society
Farber M
(2007)

*Homology of planar polygon spaces*in Geometriae Dedicata