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Persistence theory. From quiver representations to data analysis. (English) Zbl 1335.55001
Mathematical Surveys and Monographs 209. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2545-6/hbk). viii, 218 p. (2015).
Suppose that $$K$$ is a topological space in Euclidean space and that $$P$$ is a sampling of points of $$K$$. The hope is that one can infer topological information about $$K$$ from $$P$$. One approach is to consider the set consisting of the union of balls of small radius centered at the points of $$P$$. One can construct a simplicial complex based on the balls and derive the homology of the complex. The good news is that, under reasonable conditions on $$K$$ the complex yields the homology of $$K$$, the bad news it that obtaining the right granularity of the point set $$P$$ is not guaranteed and further the amount of computation required grows rapidly with the number of points in $$P$$. The issue of granularity can be addressed by the use of persistence theory. The issue of finding a way to construct complexes from $$P$$ that reduce the computational complexity is an area of active research. The aim of this book is to give an introduction to the use of persistence theory as a means of data analysis. The foregoing is a high level summary of the book’s message.
The book is divided into three parts, a theoretical introduction to persistence theory from both an algebraic viewpoint and a topological one; the second part of the book deals with applications; and the third called perspectives looks at the trends in topological data analysis and future prospects. There is an appendix on quiver theory oriented towards persistence which includes the proofs of the main results in the theory. There is a brief introduction and a comprehensive bibliography.
The book would be an excellent place to start for anyone wishing to learn about the use of persistence theory in the study of topological data analysis.

##### MSC:
 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 55N35 Other homology theories in algebraic topology 55U10 Simplicial sets and complexes in algebraic topology 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68W30 Symbolic computation and algebraic computation 68-02 Research exposition (monographs, survey articles) pertaining to computer science 62R40 Topological data analysis