Homology theories in topological data analysis

在现代拓扑数据分析的中心，人们发现了持久同调，它提供了一些基本的可计算不变量，这些不变量已经在各种各样的上下文中被发现使用。这个项目将首先集中于开发持久同源的形式性质，从切除性质开始。这里的目的是完成已知的持久同调的形式性质列表，使其与其他已知的同调理论相似。该项目的第二个目标是为简单配合物发展一种新的扭曲德拉姆型同调理论。这将把Brodzki， Mukherjee和Gao的工作从图的情况扩展到一般情况。这将提供一个非常有用的代数和几何结构的概括，是在同步问题的中心。该项目还将调查这些想法的适当实际应用。

At the centre of modern topological data analysis one finds persistent homology, which provides some of the essential computable invariants that have found use in a wide variety of contexts. This project will focus first of all on developing formal properties of persistent homology, starting with the excision property. The aim here is to complete the list of already known formal properties of persistent homology to make it similar to other known homology theories. A second aim of the project is to develop a new form of twisted de Rham-type homology theory for simplicial complexes. This will extend the work of Brodzki, Mukherjee and Gao from the case of graphs to the general case. This will provide a very useful generalisation of the algebraic and geometric structure that is at the centre of synchronisation problems. The project will also investigate appropriate practical applications of these ideas.