借助于组合结构开展空间拓扑性质的研究和不变量的计算是代数拓扑的重要内容。该思想可以追溯到代数拓扑的发展初期，在十九世纪末，庞加莱提出流形的三角剖分、单纯复形等工具，借助它们证明了流形的同调对偶定理。在当代以计算机为工具对图形图像处理及大数据分析的研究中，也以组合结构为基础发展了新的计算拓扑理论。在实际应用中，恰当的组合结构对于对象的计算机表示、分析、处理和算法设计提供了重要的帮助。为此，本项目拟开展以下几方面的研究：.(1)从组合代数拓扑角度分析数字图像的局部组合结构与相关拓扑性质；.(2)一类具有特殊组合结构的多面体复形的拓扑性质研究及其在数字图像分析中的应用；.(3)探讨基于组合结构和拓扑结构的数据分类与聚类，同时以多面体复形的持续逼近及持续同调为工具，探究大数据的整体形状结构，实现将组合代数拓扑的理论和方法用于机器学习和大数据分析。

It is an important part of the algebraic topology to study topological properties of spaces and to compute invariants by means of combinatorial structures of spaces. This idea can be traced back to the early stage of the development of algebraic topology, at the end of nineteenth century, Poincare proposed the notion of triangulations and simplicial complexes, which could be used to prove the homological duality theorem for manifolds. With the development of computer applications to the image processing and the big data analysis, a computational topology theory based on the combinatorial structure has been gradually developed. In applicaion,the appropriate combination structure of the object is helpful for its computer representation, analysis, processing and algorithm design. Therefore, this project focuses on the following subjects:.(1) From the point of view of combinatorial algebraic topology,discussing locally combinatorial structures of digital images and their topological properties; .(2) Exploring topological properties of polyhedral complex with a specific combinatorial structure and applications to digital image analysis; .(3) Discussing the data classification and the data clustering based on combinatorial structures and topology, exploring global shape of the big data by means of persistent approximation and persistent homology of polyhedral complex, and applying the combinatorial algebraic topology to the machine learning and the big data analysis.