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Author:   J. M. Landsberg
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Volume:   128
ISBN:  

9781470479053

ISBN 10:   1470479052
Pages:   439
Publication Date:   31 October 2021
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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 Part 4. Advanced topics 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. -- MAA Reviews

Author Information

J. M. Landsberg, Texas A&M University, College Station, TX

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