Applications of Algebraic Topology

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.