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Tensors: Geometry and Applications

J. M. Landsberg Texas A&M University, College Station, TX
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Hardcover ISBN: 978-0-8218-6907-9
Product Code: GSM/128
List Price: $83.00 MAA Member Price:$74.70
AMS Member Price: $66.40 Electronic ISBN: 978-0-8218-8483-6 Product Code: GSM/128.E List Price:$78.00
MAA Member Price: $70.20 AMS Member Price:$62.40
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List Price: $124.50 MAA Member Price:$112.05
AMS Member Price: $99.60 Click above image for expanded view Tensors: Geometry and Applications J. M. Landsberg Texas A&M University, College Station, TX Available Formats:  Hardcover ISBN: 978-0-8218-6907-9 Product Code: GSM/128  List Price:$83.00 MAA Member Price: $74.70 AMS Member Price:$66.40
 Electronic ISBN: 978-0-8218-8483-6 Product Code: GSM/128.E
 List Price: $78.00 MAA Member Price:$70.20 AMS Member Price: $62.40 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:$124.50 MAA Member Price: $112.05 AMS Member Price:$99.60
• Book Details

Volume: 1282012; 439 pp
MSC: Primary 15; 68; 14; 94; 20; 62;

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language.

This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Graduate students and research mathematicians interested in tensors; researchers in the sciences and geometry.

• Part 1. Motivation from applications, multilinear algebra and elementary results
• Chapter 1. Introduction
• Chapter 2. Multilinear algebra
• Chapter 3. Elementary results on rank and border rank
• Part 2. Geometry and representation theory
• Chapter 4. Algebraic geometry for spaces of tensors
• Chapter 5. Secant varieties
• Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors
• Chapter 7. Tests for border rank: Equations for secant varieties
• Chapter 8. Additional varieties useful for spaces of tensors
• Chapter 9. Rank
• Chapter 10. Normal forms for small tensors
• Part 3. Applications
• Chapter 11. The complexity of matrix multiplication
• Chapter 12. Tensor decomposition
• Chapter 13. $\mathbf {P}$ v. $\mathbf {NP}$
• Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics
• Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem
• Chapter 16. Representation theory
• Chapter 17. Weyman’s method
• Hints and answers to selected exercises

• Reviews

• I am no specialist on this subject, so I found Tensors difficult but fascinating. ...The exposition is terse, very much in the style of a graduate textbook. The reader must work through the book and become conversant with the subject. ... Most readers will enjoy the preface and chapter 1, which set out the main problems and the motivation from applied mathematics. ...A reader who knows linear and multilinear algebra and wants to know more about these questions could read Part 1 with profit. Part 2 is where the real work is done, with algebraic geometry and representation theory being the main tools. The text gets significantly denser. There is a lot of mathematics here, enough for a graduate course on this material. Part 3 returns to the applications and puts the theory to use. Part 4 is a kind of supplement that gives proofs that require more advanced techniques and discusses other advanced topics.

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Volume: 1282012; 439 pp
MSC: Primary 15; 68; 14; 94; 20; 62;

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language.

This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Graduate students and research mathematicians interested in tensors; researchers in the sciences and geometry.

• Part 1. Motivation from applications, multilinear algebra and elementary results
• Chapter 1. Introduction
• Chapter 2. Multilinear algebra
• Chapter 3. Elementary results on rank and border rank
• Part 2. Geometry and representation theory
• Chapter 4. Algebraic geometry for spaces of tensors
• Chapter 5. Secant varieties
• Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors
• Chapter 7. Tests for border rank: Equations for secant varieties
• Chapter 8. Additional varieties useful for spaces of tensors
• Chapter 9. Rank
• Chapter 10. Normal forms for small tensors
• Part 3. Applications
• Chapter 11. The complexity of matrix multiplication
• Chapter 12. Tensor decomposition
• Chapter 13. $\mathbf {P}$ v. $\mathbf {NP}$
• Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics