组合对象上持续同调的稳定性

The stability of persistent homology is the foundation for the applications of persistent homology in data analysis. In this project, we study the stability of the persistent homology for combinatorial objects, such as graphs, digraphs, hypergraphs, and simplicial complexes. We generalize the persistent homology theory, study the persistent homomorphisms between persistent homology, and construct the persistent diagrams for such persistent homomorphisms. By applying the persistent diagrams of persistent homomorphisms, we study the stability of the persistent homomorphisms between persistent homology induced by the morphisms between graphs, digraphs, hypergraphs, and simplicial complexes.

持续同调的稳定性，是将持续同调应用到数据分析问题上的理论基础。本项目中，我们研究，图、有向图、超图、单纯复形等组合对象上，持续同调的稳定性。我们将持续同调的理论做推广，研究持续同调之间的持续同态，并构造这样的持续同态的持续图表。我们利用持续同态的持续图表，研究图、有向图、超图、单纯复形等组合对象的态射所诱导的持续同调的持续同态的稳定性。

此项目中，我们主要研究了超图上的持续同调的稳定性、单纯复形的权重持续同调、有向图的道路同调等内容。其科学意义是：为超图、单纯复形、有向图的复杂网络的持续同调方法提供了数学基础。