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The Structure and Stability of Persistence Modules

2016-10-08 Mathematics
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Release : 2016-10-08
Genre : Mathematics
Kind : eBook
Book Rating : 455/5 ( reviews)

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Book Synopsis The Structure and Stability of Persistence Modules by : Frédéric Chazal

Download or read book The Structure and Stability of Persistence Modules written by Frédéric Chazal. This book was released on 2016-10-08. Available in PDF, EPUB and Kindle. Book excerpt: This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such modules, providing a sound mathematical framework for the study of persistence diagrams. Completely self-contained, this brief introduces the notion of persistence measure and makes extensive use of a new calculus of quiver representations to facilitate explicit computations. Appealing to both beginners and experts in the subject, The Structure and Stability of Persistence Modules provides a purely algebraic presentation of persistence, and thus complements the existing literature, which focuses mainly on topological and algorithmic aspects.

The Structure and Stability of Persistence Modules

2017-06-07
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Release : 2017-06-07
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Kind : eBook
Book Rating : 010/5 ( reviews)

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Book Synopsis The Structure and Stability of Persistence Modules by : Jack Noah

Download or read book The Structure and Stability of Persistence Modules written by Jack Noah. This book was released on 2017-06-07. Available in PDF, EPUB and Kindle. Book excerpt: This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such modules, providing a sound mathematical framework for the study of persistence diagrams. Completely self-contained, this brief introduces the notion of persistence measure and makes extensive use of a new calculus of quiver representations to facilitate explicit computations. Appealing to both beginners and experts in the subject, The Structure and Stability of Persistence Modules provides a purely algebraic presentation of persistence, and thus complements the existing literature, which focuses mainly on topological and algorithmic aspect

Persistence Theory: From Quiver Representations to Data Analysis

2017-05-17 Mathematics
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Release : 2017-05-17
Genre : Mathematics
Kind : eBook
Book Rating : 431/5 ( reviews)

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Book Synopsis Persistence Theory: From Quiver Representations to Data Analysis by : Steve Y. Oudot

Download or read book Persistence Theory: From Quiver Representations to Data Analysis written by Steve Y. Oudot. This book was released on 2017-05-17. Available in PDF, EPUB and Kindle. Book excerpt: Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work. The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.

Topological Persistence in Geometry and Analysis

2020-05-11 Education
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Release : 2020-05-11
Genre : Education
Kind : eBook
Book Rating : 955/5 ( reviews)

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Book Synopsis Topological Persistence in Geometry and Analysis by : Leonid Polterovich

Download or read book Topological Persistence in Geometry and Analysis written by Leonid Polterovich. This book was released on 2020-05-11. Available in PDF, EPUB and Kindle. Book excerpt: The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology.

Quantitative Tamarkin Theory

2020-03-09 Mathematics
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Release : 2020-03-09
Genre : Mathematics
Kind : eBook
Book Rating : 888/5 ( reviews)

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Book Synopsis Quantitative Tamarkin Theory by : Jun Zhang

Download or read book Quantitative Tamarkin Theory written by Jun Zhang. This book was released on 2020-03-09. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers readers a self-contained introduction to quantitative Tamarkin category theory. Functioning as a viable alternative to the standard algebraic analysis method, the categorical approach explored in this book makes microlocal sheaf theory accessible to a wide audience of readers interested in symplectic geometry. Much of this material has, until now, been scattered throughout the existing literature; this text finally collects that information into one convenient volume. After providing an overview of symplectic geometry, ranging from its background to modern developments, the author reviews the preliminaries with precision. This refresher ensures readers are prepared for the thorough exploration of the Tamarkin category that follows. A variety of applications appear throughout, such as sheaf quantization, sheaf interleaving distance, and sheaf barcodes from projectors. An appendix offers additional perspectives by highlighting further useful topics. Quantitative Tamarkin Theory is ideal for graduate students interested in symplectic geometry who seek an accessible alternative to the algebraic analysis method. A background in algebra and differential geometry is recommended. This book is part of the "Virtual Series on Symplectic Geometry" http://www.springer.com/series/16019

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