Softcover ISBN:  9781470464554 
Product Code:  GSM/207.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Electronic ISBN:  9781470460082 
Product Code:  GSM/207.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 

Book DetailsGraduate Studies in MathematicsVolume: 207; 2020; 312 ppMSC: 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 standalone 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.ReadershipGraduate students and researchers interested in applied topology.

Table of Contents

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


Additional Material

Reviews

The book is wellequipped 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 VejdemoJohansson, CUNY College of Staten Island


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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 standalone 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 wellequipped 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 VejdemoJohansson, CUNY College of Staten Island