Applications of Algebraic Topology

在这项研究中，我们将研究由应用启发的代数拓扑问题。考虑一个机械系统，其功能如下：操作员引入系统的当前状态和期望状态，系统的计算机（使用先验设计的运动规划算法）确定系统从当前状态到期望状态的连续运动。在本研究中，我们研究了这种运动规划算法的拓扑性质。在大多数情况下，这种运动规划算法是不连续的，即输入数据的微小扰动可能导致运动曲线的显著变化。我们计划在本研究中进一步发展的理论允许使用代数拓扑方法来描述运动规划算法中的不稳定性并最小化它们。我们还将处理由机器人应用程序启发的其他几个有趣的拓扑问题。

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.