1. TJ'he first step toward connecting these two basic concepts of topology was taken by L. E. J. Brouwer in 1912 by demonstrating that two continuous mappings of a two-dimensional sphere into itself can be continuously deformed into each other if and only if they have the same degree (that is, if they are equivalent from the point of view of homology theory). After having generalized Brouwer's ...

This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using mathematical concept topology. We explain fundamental idea homology from scratch some examples. also review applications materials researches and software for analysis. HomCloud, one software, especially featured in this paper.

In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds torus boundary in terms immersed curves a punctured torus. particular, knot is captured by this invariant. Appealing to earlier the authors on bordered homology, give formula behaviour these under cabling.