Tools of Applied Algebraic Topology

该项目开发了与各种应用相关的代数拓扑工具。我们将研究构建运动规划算法问题的拓扑方面以及反映这些算法复杂性的拓扑空间的同伦不变量。这些不变量是在前面的一些有趣的例子中计算出来的，在本研究中，我们计划解决几个剩下的重要挑战，并将结果用于许多应用程序。我们计划用空间X的基本群来表示非球面空间X的拓扑复杂度TC(X)，希望找到与同调代数和几何群论的新联系。我们计划进一步探索运动规划问题与经典几何拓扑问题（流形的嵌入和浸入）在透镜空间中的联系，希望找到关于实射影空间的最新结果的推广。我们还计划研究运动规划算法理论的修改及其与并发计算理论相关的拓扑复杂性。在本研究中，我们还将研究处理依赖大量随机参数的拓扑空间的混合拓扑-概率特征的新问题。研究此类空间的主要动机来自涉及大型机械系统的工程应用；它们的构形空间是这种随机拓扑空间的主要例子。在我们以前的工作中，我们详细研究了具有大量连杆和随机杆长的机械连杆的构型空间。在这项研究中，我们计划研究产生随机拓扑空间的其他机制，例如随机大小的粒子的配置空间，以及基于随机图的各种模型。

The project develops tools of algebraic topology which are relevant to various applications. We will study topological aspects of the problem of building motion planning algorithms and homotopy invariants of topological spaces which reflect the complexity of these algorithms. These invariants were computed previously for some interesting examples, and in this research we plan to resolve several remaining important challenges and also adopt the results for a number of applications. We plan to express the topological complexity TC(X) for aspherical spaces X in terms of the fundamental group of the space X hoping to find new connections with homological algebra and geometric group theory. We plan to explore further the connection between the motion planning problem and the classical problems of geometric topology (embeddings and immersions of manifolds) in the case of lens spaces hoping to find generalizations of recent results concerning real projective spaces. We also plan to study a modification of the theory of motion planning algorithms and their topological complexity relevant to the theory of concurrent computation. In this research we will also study new problems of a mixed topological - probabilistic character dealing with topological spaces depending on a large number of random parameters. The main motivation for studying such spaces comes from engineering applications involving large mechanical systems; their configuration spaces are major examples of such random topological spaces . In our previous work we studied in detail configuration spaces of mechanical linkages with large number of links and with random bar lengths. In this research we plan to investigate other mechanisms producing random topological spaces, such as configuration spaces of particles of random size, and various models based on random graphs.