论文标题
多参数离散理论
Multiparameter Discrete Morse Theory
论文作者
论文摘要
本文的主要目的是将摩尔斯 - 格式理论扩展到矢量值函数。这主要是由于需要开发新工具和方法来计算多参数持久性的动机。为了概括该理论，除了将Forman的主要定义和结果调整为此矢量环境外，我们还使用了近年来研究的组合拓扑动力学概念。这种方法被证明是成功的。首先，我们建立了一个比福尔曼在多维离散摩尔斯函数的级别集的结果。其次，我们找到了一种方法，可以从这种功能的临界点中诱导关键组件中的摩尔斯分解。最后，我们推断出一组摩尔斯方程和多参数设置特定的不平等现象。
The main objective of this paper is to extend Morse-Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition to adapting the main definitions and results of Forman to this vectorial setting, we use concepts of combinatorial topological dynamics studied in recent years. This approach proves to be successful in the following ways. First, we establish a result which is more general than that of Forman regarding the sublevel sets of a multidimensional discrete Morse function. Second, we find a way to induce a Morse decomposition in critical components from the critical points of such a function. Finally, we deduce a set of Morse equation and inequalities specific to the multiparameter setting.
