** 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.

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

*Preface*

0.1. Usage

0.2. Overview

0.3. Clash of cultures

0.4. Further reading

0.5. Conventions, acknowledgments

**Part 1. Motivation from applications, multilinear algebra, and elementary results**

Chapter 1. Introduction

Chapter 1. Introduction

1.1. The complexity of matrix multiplication

1.2. Definitions from multilinear algebra

1.3. Tensor decomposition

1.4. P v. NP and algebraic variants

1.5. Algebraic statistics and tensor networks

1.6. Geometry and representation theory

**Chapter 2. Multilinear algebra**

2.1. Rust removal exercises

2.2. Groups and representations

2.3. Tensor products

2.4. The rank and border rank of a tensor

2.5. Examples of invariant tensors

2.6. Symmetric and skew-symmetric tensors

2.7. Polynomials on the space of matrices

2.8. Decomposition of V ?3

2.9. Appendix: Basic definitions from algebra

2.10. Appendix: Jordan and rational canonical form

2.11. Appendix: Wiring diagrams

**Chapter 3. Elementary results on rank and border rank**

3.1. Ranks of tensors

3.2. Symmetric rank

3.3. Uniqueness of CP decompositions

3.4. First tests of border rank: flattenings

3.5. Symmetric border rank

3.6. Partially symmetric tensor rank and border rank

3.7. Two useful techniques for determining border rank

3.8. Strassen's equations and variants

3.9. Equations for small secant varieties

3.10. Equations for symmetric border rank

3.11. Tensors in C2?Cb?Cc

**Part 2. Geometry and representation theory**

Chapter 4. Algebraic geometry for spaces of tensors

Chapter 4. Algebraic geometry for spaces of tensors

4.1. Diagnostic test for those familiar with algebraic geometry

4.2. First definitions

4.3. Examples of algebraic varieties

4.4. Defining equations of Veronese re-embeddings

4.5. Grassmannians

4.6. Tangent and cotangent spaces to varieties

4.7. G-varieties and homogeneous varieties

4.8. Exercises on Jordan normal form and geometry

4.9. Further information regarding algebraic varieties

**Chapter 5. Secant varieties**

5.1. Joins and secant varieties

5.2. Geometry of rank and border rank

5.3. Terracini's lemma and first consequences

5.4. The polynomial Waring problem

5.5. Dimensions of secant varieties of Segre varieties

5.6. Ideas of proofs of dimensions of secant varieties for triple Segre products

5.7. BRPP and conjectures of Strassen and Comon

**Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors**

6.1. Schur's lemma

6.2. Finite groups

6.3. Representations of the permutation group Sd

6.4. Decomposing V ?d as a GL(V )-module with the aid of Sd

6.5. Decomposing Sd(A1? · · · ? An) as a G = GL(A1) × · · · ×

GL(An)-module

6.6. Characters

6.7. The Littlewood-Richardson rule

6.8. Weights and weight spaces: a generalization of eigenvalues and eigenspaces

6.9. Homogeneous varieties

6.10. Ideals of homogeneous varieties

6.11. Symmetric functions

**Chapter 7. Tests for border rank: Equations for secant varieties**

7.1. Subspace varieties and multilinear rank

7.2. Additional auxiliary varieties

7.3. Flattenings

7.4. Inheritance

7.5. Prolongation and multiprolongation

7.6. Strassen's equations, applications and generalizations

7.7. Equations for s4(Seg(PA × PB × PC))

7.8. Young flattenings

**Chapter 8. Additional varieties useful for spaces of tensors**

8.1. Tangential varieties

8.2. Dual varieties

8.3. The Pascal determinant

8.4. Differential invariants of projective varieties

8.5. Stratifications of PV * via dual varieties

8.6. The Chow variety of zero cycles and its equations

8.7. The Fano variety of linear spaces on a variety

**Chapter 9. Rank**

9.1. Remarks on rank for arbitrary varieties

9.2. Bounds on symmetric rank

9.3. Examples of classes of polynomials and their ranks

**Chapter 10. Normal forms for small tensors**

10.1. Vector spaces with a finite number of orbits

10.2. Vector spaces where the orbits can be explicitly parametrized

10.3. Points in C2?Cb?Cc

10.4. Ranks and border ranks of elements of S3C3

10.5. Tensors in C3?C3?C3

10.6. Normal forms for C2?S2W

10.7. Exercises on normal forms for general points on small secant varieties

10.8. Limits of secant planes

10.9. Limits for Veronese varieties

10.10. Ranks and normal forms in s3(Seg(PA1? · · ·? PAn))

**Part 3. Applications**

**Chapter 11. The complexity of matrix multiplication**

11.1. "Real world" issues

11.2. Failure of the border rank version of Strassen's conjecture

11.3. Finite group approach to upper bounds

11.4. R(M3,3,3) = 23

11.5. Bl¨aser's 5

2-Theorem

11.6. The Brockett-Dobkin Theorem

11.7. Multiplicative complexity

**Chapter 12. Tensor decomposition**

12.1. Cumulants

12.2. Blind deconvolution of DS-CMDA signals

12.3. Uniqueness results coming from algebraic geometry

12.4. Exact decomposition algorithms

12.5. Kruskal's theorem and its proof

**Chapter 13. P v. NP**

13.1. Introduction to complexity

13.2. Polynomials in complexity theory, graph theory, and statistics

13.3. Definitions of VP, VNP, and other algebraic complexity classes

13.4. Complexity of permn and detn

13.5. Immanants and their symmetries

13.6. Geometric complexity theory approach to VPws v. VNP

13.7. Other complexity classes via polynomials

13.8. Vectors of minors and homogeneous varieties

13.9. Holographic algorithms and spinors

Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics

Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics

14.1. Tensor network states

14.2. Algebraic statistics and phylogenetics

Part 4. Advanced topics

Part 4. Advanced topics

**Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem**

15.1. The semiclassical cases

15.2. The Alexander-Hirschowitz idea for dealing with the remaining cases

Chapter 16. Representation theory

Chapter 16. Representation theory

16.1. Basic definitions

16.2. Casimir eigenvalues and Kostant's theorem

16.3. Cohomology of homogeneous vector bundles

16.4. Equations and inheritance in a more general context

Chapter 17. Weyman's method

Chapter 17. Weyman's method

17.1. Ideals and coordinate rings of projective varieties

17.2. Koszul sequences

17.3. The Kempf-Weyman method

17.4. Subspace varieties

Hints and answers to selected exercises

Bibliography

*Index*