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Organized Collapse: An Introduction to Discrete Morse Theory

Dmitry N. Kozlov University of Bremen, Bremen, Germany and Okinawa Institute of Science and Technology, Okinawa, Japan
Available Formats:
Softcover ISBN: 978-1-4704-6455-4
Product Code: GSM/207.S
List Price: $89.00 MAA Member Price:$80.10
AMS Member Price: $71.20 Electronic ISBN: 978-1-4704-6008-2 Product Code: GSM/207.E List Price:$89.00
MAA Member Price: $80.10 AMS Member Price:$71.20
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List Price: $133.50 MAA Member Price:$120.15
AMS Member Price: $106.80 Click above image for expanded view Organized Collapse: An Introduction to Discrete Morse Theory Dmitry N. Kozlov University of Bremen, Bremen, Germany and Okinawa Institute of Science and Technology, Okinawa, Japan Available Formats:  Softcover ISBN: 978-1-4704-6455-4 Product Code: GSM/207.S  List Price:$89.00 MAA Member Price: $80.10 AMS Member Price:$71.20
 Electronic ISBN: 978-1-4704-6008-2 Product Code: GSM/207.E
 List Price: $89.00 MAA Member Price:$80.10 AMS Member Price: $71.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$133.50 MAA Member Price: $120.15 AMS Member Price:$106.80
• Book Details

Volume: 2072020; 312 pp
MSC: Primary 57; Secondary 05; 06; 55; 58;

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.

Graduate students and researchers interested in applied topology.

• Introduction to homology
• The first steps
• Simplicial homology
• Beyond the simplicial setting
• Further aspects of homology theory
• Category of chain complexes
• Chain homotopy
• Connecting homomorphism
• Singular homology
• Cellular homology
• Suggested further reading for parts 1 and 2
• Basic discrete Morse theory
• Simplicial collapses
• Organizing collapsing sequences
• Internal collapses and discrete Morse theory
• Explicit homology classes associated to critical cells
• The critical Morse complex
• Implications and variations
• Suggested further reading for part 3
• Extensions of discrete Morse theory
• Algebraic Morse theory
• Discrete Morse theory for posets
• Discrete Morse theory for CW complexes
• Disctrete Morse theory and persistence
• Suggested further reading for part 4

• Reviews

• The book is well-equipped with both illustrative examples, many of them drawing on combinatorics and on graph theory, and plenty of exercises gathered at the end of each chapter. In each of the four parts of the book, suggestions for further reading are included with comments guiding a reader to a targeted exploration of the literature. I expect it to find regular use as a reference myself.

Mikael Vejdemo-Johansson, CUNY College of Staten Island
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2072020; 312 pp
MSC: Primary 57; Secondary 05; 06; 55; 58;

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.

Graduate students and researchers interested in applied topology.

• Introduction to homology
• The first steps
• Simplicial homology
• Beyond the simplicial setting
• Further aspects of homology theory
• Category of chain complexes
• Chain homotopy
• Connecting homomorphism
• Singular homology
• Cellular homology
• Suggested further reading for parts 1 and 2
• Basic discrete Morse theory
• Simplicial collapses
• Organizing collapsing sequences
• Internal collapses and discrete Morse theory
• Explicit homology classes associated to critical cells
• The critical Morse complex
• Implications and variations
• Suggested further reading for part 3
• Extensions of discrete Morse theory
• Algebraic Morse theory
• Discrete Morse theory for posets
• Discrete Morse theory for CW complexes
• Disctrete Morse theory and persistence
• Suggested further reading for part 4
• The book is well-equipped with both illustrative examples, many of them drawing on combinatorics and on graph theory, and plenty of exercises gathered at the end of each chapter. In each of the four parts of the book, suggestions for further reading are included with comments guiding a reader to a targeted exploration of the literature. I expect it to find regular use as a reference myself.

Mikael Vejdemo-Johansson, CUNY College of Staten Island
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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