Discrete Morse Theory

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Discrete Morse Theory

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Author : Nicholas A. Scoville
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-09-27

Discrete Morse Theory written by Nicholas A. Scoville and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-27 with Mathematics categories.

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.

Morse Theory Smooth And Discrete

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Author : Kevin P Knudson
language : en
Publisher: World Scientific Publishing Company
Release Date : 2015-05-29

Morse Theory Smooth And Discrete written by Kevin P Knudson and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-29 with Mathematics categories.

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.

An Invitation To Morse Theory

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Author : Liviu Nicolaescu
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-02

An Invitation To Morse Theory written by Liviu Nicolaescu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-02 with Mathematics categories.

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.

Morse Homology

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Author : Schwarz
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Morse Homology written by Schwarz and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

1.1 Background The subject of this book is Morse homology as a combination of relative Morse theory and Conley's continuation principle. The latter will be useda s an instrument to express the homology encoded in a Morse complex associated to a fixed Morse function independent of this function. Originally, this type of Morse-theoretical tool was developed by Andreas Floer in order to find a proof of the famous Arnold conjecture, whereas classical Morse theory turned out to fail in the infinite-dimensional setting. In this framework, the homological variant of Morse theory is also known as Floer homology. This kind of homology theory is the central topic of this book. But first, it seems worthwhile to outline the standard Morse theory. 1.1.1 Classical Morse Theory The fact that Morse theory can be formulated in a homological way is by no means a new idea. The reader is referred to the excellent survey paper by Raoul Bott [Bol.

An Introduction To Morse Theory

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Author : Yukio Matsumoto
language : en
Publisher: American Mathematical Soc.
Release Date : 2002

An Introduction To Morse Theory written by Yukio Matsumoto and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.

This book introduces basic concepts related to finite dimensions, including critical points, the Hessian, and handle decompressions. It first uses surfaces to illustrate these ideas, and then generalizes them to apply to higher dimensions. This treatment then informs a discussion of handlebodies, homology, and low-dimensional manifold theory. Illustrations are provided throughout. c. Book News Inc.

Morse Theory Am 51 Volume 51

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Author : John Milnor
language : en
Publisher: Princeton University Press
Release Date : 2016-03-02

Morse Theory Am 51 Volume 51 written by John Milnor and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-02 with Mathematics categories.

One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory. Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.? Milnor has published five books with Princeton University Press.

Stratified Morse Theory

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Author : Mark Goresky
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Stratified Morse Theory written by Mark Goresky and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

Simplicial Complexes Of Graphs

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Author : Jakob Jonsson
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-11-15

Simplicial Complexes Of Graphs written by Jakob Jonsson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-11-15 with Mathematics categories.

A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory.

Computational Topology For Data Analysis

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Author : Tamal Krishna Dey
language : en
Publisher: Cambridge University Press
Release Date : 2022-03-10

Computational Topology For Data Analysis written by Tamal Krishna Dey and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-10 with Computers categories.

This book provides a computational and algorithmic foundation for techniques in topological data analysis, with examples and exercises.