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Discrete Morse Theory

2019-09-27 Combinatorial topology
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Release : 2019-09-27
Genre : Combinatorial topology
Kind : eBook
Book Rating : 987/5 ( reviews)

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Book Synopsis Discrete Morse Theory by : Nicholas A. Scoville

Download or read book Discrete Morse Theory written by Nicholas A. Scoville. This book was released on 2019-09-27. Available in PDF, EPUB and Kindle. Book excerpt: Discrete Morse theory is a powerful tool combining ideas in both topology and combinatorics. Invented by Robin Forman in the mid 1990s, discrete Morse theory is a combinatorial analogue of Marston Morse's classical Morse theory. Its applications are vast, including applications to topological data analysis, combinatorics, and computer science. This book, the first one devoted solely to discrete Morse theory, serves as an introduction to the subject. Since the book restricts the study of discrete Morse theory to abstract simplicial complexes, a course in mathematical proof writing is the only prerequisite needed. Topics covered include simplicial complexes, simple homotopy, collapsibility, gradient vector fields, Hasse diagrams, simplicial homology, persistent homology, discrete Morse inequalities, the Morse complex, discrete Morse homology, and strong discrete Morse functions. Students of computer science will also find the book beneficial as it includes topics such as Boolean functions, evasiveness, and has a chapter devoted to some computational aspects of discrete Morse theory. The book is appropriate for a course in discrete Morse theory, a supplemental text to a course in algebraic topology or topological combinatorics, or an independent study.

Organized Collapse: An Introduction to Discrete Morse Theory

2021-02-18 Mathematics
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Release : 2021-02-18
Genre : Mathematics
Kind : eBook
Book Rating : 551/5 ( reviews)

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Book Synopsis Organized Collapse: An Introduction to Discrete Morse Theory by : Dmitry N. Kozlov

Download or read book Organized Collapse: An Introduction to Discrete Morse Theory written by Dmitry N. Kozlov. This book was released on 2021-02-18. Available in PDF, EPUB and Kindle. Book excerpt: Applied topology is a modern subject which emerged in recent years at a crossroads of many methods, all of them topological in nature, which were used in a wide variety of applications in classical mathematics and beyond. Within applied topology, discrete Morse theory came into light as one of the main tools to understand cell complexes arising in different contexts, as well as to reduce the complexity of homology calculations. The present book provides a gentle introduction into this beautiful theory. Using a combinatorial approach—the author emphasizes acyclic matchings as the central object of study. The first two parts of the book can be used as a stand-alone introduction to homology, the last two parts delve into the core of discrete Morse theory. The presentation is broad, ranging from abstract topics, such as formulation of the entire theory using poset maps with small fibers, to heavily computational aspects, providing, for example, a specific algorithm of finding an explicit homology basis starting from an acyclic matching. The book will be appreciated by graduate students in applied topology, students and specialists in computer science and engineering, as well as research mathematicians interested in learning about the subject and applying it in context of their fields.

Morse Theory: Smooth And Discrete

2015-05-29 Mathematics
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Release : 2015-05-29
Genre : Mathematics
Kind : eBook
Book Rating : 985/5 ( reviews)

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Book Synopsis Morse Theory: Smooth And Discrete by : Kevin P Knudson

Download or read book Morse Theory: Smooth And Discrete written by Kevin P Knudson. This book was released on 2015-05-29. Available in PDF, EPUB and Kindle. Book excerpt: Morse Theory: Smooth and Discrete serves as an introduction to classical smooth Morse theory and to Forman's discrete Morse theory, highlighting the parallels between the two subjects. This is the first time both smooth and discrete Morse theory have been treated in a single volume. This makes the book a valuable resource for students and professionals working in topology and discrete mathematics. With a strong focus on examples, the text is suitable for advanced undergraduates or beginning graduate students.

Stratified Morse Theory

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 144/5 ( reviews)

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Book Synopsis Stratified Morse Theory by : Mark Goresky

Download or read book Stratified Morse Theory written by Mark Goresky. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.

An Invitation to Morse Theory

2011-12-02 Mathematics
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Release : 2011-12-02
Genre : Mathematics
Kind : eBook
Book Rating : 05X/5 ( reviews)

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Book Synopsis An Invitation to Morse Theory by : Liviu Nicolaescu

Download or read book An Invitation to Morse Theory written by Liviu Nicolaescu. This book was released on 2011-12-02. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained treatment of Morse theory focuses on applications and is intended for a graduate course on differential or algebraic topology, and will also be of interest to researchers. This is the first textbook to include topics such as Morse-Smale flows, Floer homology, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The reader is expected to have some familiarity with cohomology theory and differential and integral calculus on smooth manifolds. Some features of the second edition include added applications, such as Morse theory and the curvature of knots, the cohomology of the moduli space of planar polygons, and the Duistermaat-Heckman formula. The second edition also includes a new chapter on Morse-Smale flows and Whitney stratifications, many new exercises, and various corrections from the first edition.

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