Topological Persistence in Geometry and Analysis

Topology has enjoyed a great resurgence in the last 20 years due in large part to the emergence of persistent homology. While much of the focus of persistence is on real world applications such as data analysis, persistence can also help shed light other areas of mathematics. As the authors note in the preface, the purpose of the book under review is to provide an introduction to the applications of persistence modules to function theory and symplectic geometry. There are few books, if any, whose main object is to apply the methods and techniques of persistence to symplectic geometry so such a text is certainly needed. The necessary prerequisites for the book are a basic background knowledge of both algebraic and differential topology, and as such, the book is primarily intended for researchers or those interested in learning more about the applications of persistence to geometry. The book is organized into three parts with each part consisting of between two and four chapters.

 

Part I introduces persistence modules, the main objects of study of Chapter 1. These are an objects of linear algebra or, what might be worse for some, categorical objects. However, we often think of a persistence module as simply a collection of intervals or barcode, a much more digestible viewpoint. After introducing the simple concept of barcodes in Chapter 2, the authors justify the above mentioned equivalence by discussing the isometry theorem, a celebrated result of Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. Applications of the isometry theorem are discussed before giving the proof in the next chapter. Part I concludes with a chapter discussing information we can learn from a barcode. One option is to put Lipschitz functional on the space of bar codes. Doing so will yield numerical invariants of other functions and metric spaces. 

 

Part II applies the basic theory developed in Part I to metric geometry and function theory. One popular application, discussed in Chapter 5, is to the Rips complex of a set of points. The ultimate goal here is given an unknown Riemannian manifold and a sampling of points on the manifold, how can we use persistence to recover information about the manifold? It is shown that under certain hypotheses, we can use persistence to at least recover the homology of the underlying manifold. Chapter 5 investigates applications to topological function theory, a theory that studies features of smooth functions on a manifold that remain invariant under the action of the diffeomorphism group. As before, we can use persistence to construct a variety of invariants. 

 

The third and final part of the book looks at persistence in symplectic geometry. After giving a crash course in symplectic geometry, the authors apply the theory of persistence to the symplectic setting. In particular, they outline a symplectic version of the stability theorem.

 

One nice feature of this short book is that it is interactive in the sense that there are exercises for the reader to work on interspersed throughout the text. The exercises of course vary in difficulty. Another feature is that in addition to a subject index, there is both a notational index as well as a name index. It provides a good introduction to anyone wishing to learn about the subject and a reference for practitioners of the subject. 

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Nick Scoville (nscoville@ursinus.edu) is an assistant professor of mathematics at Ursinus College. His areas of interest are homotopy theory, discrete topology, and the history of topology. He considers himself an amateur scholastic. His website can be found at http://webpages.ursinus.edu/nscoville/