Primoz Skraba (2/28/18): An approximate nerve theorem скачать в хорошем качестве

Primoz Skraba (2/28/18): An approximate nerve theorem

The Nerve Theorem is an implicit tool in most applications of topological data analysis relating the topological type of a suitably nice space with a combinatorial description of the space, namely, the nerve of a cover of that space. It is required that it is a good cover, that each element and intersection is contractible or at least acyclic. In this talk, I will describe a weaker condition we call an epsilon-acyclic cover. It encodes the idea that if a cover is almost a good cover, the persistent homology of a filtration computed on the nerve is a good approximation of the persistent homology of a filtration on the underlying space. In proving the result, we derive a refined notion of interleaving and the proof also provides a way of estimating the epsilon in certain cases.