联系矩阵理论的计算框架
A computational framework for connection matrix theory
摘要：
连接矩阵是一个强大的代数拓扑工具，从康利指数理论，捕捉孤立的不变集之间的关系。Conley指数理论是莫尔斯理论的拓扑推广，其中连接矩阵包含莫尔斯边界算子的作用。在过去的几十年里，康利的思想已经被转换成一种纯粹的计算形式。在本文中，我们介绍了一个计算，分类框架的连接矩阵理论。这一贡献将计算Conley理论转化为动力系统的计算同调理论。更具体地说，在本文中，我们有两个目标：1）我们把连接矩阵理论转换成适当的范畴，同伦理论的语言。我们确定对象的适当类别，对应于连接矩阵，并可以计算内的计算康利理论范式，通过使用技术的减少。2)我们描述了一种算法，这种计算的基础上代数离散莫尔斯理论。
英文摘要：
The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.
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