[PDF] Topological Persistence In Geometry And Analysis eBook

Author : Leonid Polterovich
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 11,41 MB
Release : 2020-05-11
Category : Education
ISBN : 1470454955

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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.

Author : Tamal Krishna Dey
Publisher : Cambridge University Press
Page : 456 pages
File Size : 46,8 MB
Release : 2022-03-10
Category : Mathematics
ISBN : 1009103199

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Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.

Author : Steve Y. Oudot
Publisher : American Mathematical Soc.
Page : 229 pages
File Size : 33,37 MB
Release : 2017-05-17
Category : Mathematics
ISBN : 1470434431

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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.

Author : Jean-Daniel Boissonnat
Publisher : Cambridge University Press
Page : 247 pages
File Size : 28,68 MB
Release : 2018-09-27
Category : Computers
ISBN : 1108419399

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A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.

Author : Robert W. Ghrist
Publisher : Createspace Independent Publishing Platform
Page : 0 pages
File Size : 50,51 MB
Release : 2014
Category : Mathematics
ISBN : 9781502880857

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This book gives an introduction to the mathematics and applications comprising the new field of applied topology. The elements of this subject are surveyed in the context of applications drawn from the biological, economic, engineering, physical, and statistical sciences.

Author : Gunnar Carlsson
Publisher : Cambridge University Press
Page : 233 pages
File Size : 12,96 MB
Release : 2021-12-16
Category : Computers
ISBN : 1108838650

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This timely text introduces topological data analysis from scratch, with detailed case studies.

Author : Herbert Edelsbrunner
Publisher : American Mathematical Society
Page : 241 pages
File Size : 41,60 MB
Release : 2022-01-31
Category : Mathematics
ISBN : 1470467690

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Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

Author : Abigail Hickok
Publisher :
Page : 0 pages
File Size : 28,40 MB
Release : 2023
Category :
ISBN :

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The fields of topological data analysis (TDA) and geometric data analysis (GDA) use algebraic topology and differential geometry to capture topological and geometric structural properties of data that are not captured by other methods in data science and machine learning. The primary tool of TDA---and one of the focuses of this dissertation---is persistent homology, which measures the connected components, holes, and higher-dimensional voids of a data set and tracks how those voids emerge and disappear at different scales. The objective of GDA is to extract new insights by considering geometric invariants of a manifold, such as curvature, rather than topological invariants. Previous studies have demonstrated the power of geometry and topology for analyzing data in complex systems, neuroscience, biology, and many other fields. In my thesis, I study both the theory and applications of topological and geometric data analysis. In the first part of the dissertation, I establish and analyze a new construction, called a "persistence diagram (PD) bundle," for doing multiparameter TDA, and I develop an algorithm to compute a certain class of PD bundles. PD bundles generalize several important constructions in TDA: vineyards, the persistent homology transform, and fibered barcodes. In the second part of the dissertation, I apply TDA to several geospatial and geospatiotemporal data sets. In the last part of the dissertation, I introduce a new method for curvature estimation in point-cloud data.

Author : Nils A. Baas
Publisher : Springer Nature
Page : 522 pages
File Size : 16,54 MB
Release : 2020-06-25
Category : Mathematics
ISBN : 3030434087

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This book gathers the proceedings of the 2018 Abel Symposium, which was held in Geiranger, Norway, on June 4-8, 2018. The symposium offered an overview of the emerging field of "Topological Data Analysis". This volume presents papers on various research directions, notably including applications in neuroscience, materials science, cancer biology, and immune response. Providing an essential snapshot of the status quo, it represents a valuable asset for practitioners and those considering entering the field.

Author : Hamish Carr
Publisher : Springer Nature
Page : 264 pages
File Size : 44,83 MB
Release : 2020-12-10
Category : Mathematics
ISBN : 3030430367

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This collection of peer-reviewed workshop papers provides comprehensive coverage of cutting-edge research into topological approaches to data analysis and visualization. It encompasses the full range of new algorithms and insights, including fast homology computation, comparative analysis of simplification techniques, and key applications in materials and medical science. The book also addresses core research challenges such as the representation of large and complex datasets, and integrating numerical methods with robust combinatorial algorithms. In keeping with the focus of the TopoInVis 2017 Workshop, the contributions reflect the latest advances in finding experimental solutions to open problems in the sector. They provide an essential snapshot of state-of-the-art research, helping researchers to keep abreast of the latest developments and providing a basis for future work. Gathering papers by some of the world’s leading experts on topological techniques, the book represents a valuable contribution to a field of growing importance, with applications in disciplines ranging from engineering to medicine.