**Graduate Studies in Mathematics**

Volume: 128;
2012;
439 pp;
Hardcover

MSC: Primary 15; 68; 14; 94; 20; 62;

**Print ISBN: 978-0-8218-6907-9
Product Code: GSM/128**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-0-8218-8483-6
Product Code: GSM/128.E**

List Price: $78.00

AMS Member Price: $62.40

MAA Member Price: $70.20

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#### Supplemental Materials

# Tensors: Geometry and Applications

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*J. M. Landsberg*

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.

#### Readership

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

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Tensors: Geometry and Applications

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface xi12 free
- Part I. Motivation from applications, multilinear algebra and elementary results 122 free
- Introduction 324
- Multilinear algebra 2748
- Elementary results on rank and border rank 6788
- Part II. Geometry and representation theory 95116
- Algebraic geometry for spaces of tensors 97118
- Secant varieties 117138
- Exploiting symmetry: Representation theory for spaces of tensors 137158
- Tests for border rank: Equations for secant varieties 173194
- Additional varieties useful for spaces of tensors 207228
- Rank 229250
- Normal forms for small tensors 243264
- Part III. Applications 273294
- The complexity of matrix multiplication 275296
- Tensor decomposition 289310
- 𝐏 v. 𝐍𝐏 311332
- Varieties of tensors in phylogenetics and quantum mechanics 357378
- Part IV. Advanced topics 371392
- Overview of the proof of the Alexander-Hirschowitz theorem 373394
- Representation theory 381402
- Weyman’s method 395416
- Hints and answers to selected exercises 409430
- Bibliography 415436
- Index 433454 free
- Back Cover Back Cover1464