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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
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
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
14.1. Tensor network states
14.2. Algebraic statistics and phylogenetics

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
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
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