Part 1. Motivation from applications, multilinear algebra and elementary results Chapter 1. Introduction 3 §1.1. The complexity of matrix multiplication 5 §1.2. Definitions from multilinear algebra 6 §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 v vi Contents §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 C 2 ⊗C b ⊗C c 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 Contents vii §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 of 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 S d §6.4. Decomposing V ⊗d as a GL(V) module with the aid of S d §6.5. Decomposing S d (A 1 ⊗ • • • ⊗ A n) as a G = GL(A 1) × • • • × GL(A n)-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 multi-linear rank §7.2. Additional auxiliary varieties §7.3. Flattenings §7.4. Inheritance §7.5. Prolongation and multi-prolongation §7.6. Strassen's equations, applications and generalizations §7.7. Equations for σ 4 (Seg(PA × PB × PC)) §7.8. Young flattenings Chapter 8. Ad...

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.

Abstract.We determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these spaces, and establish basic facts about the local differential geometry of rational homogeneous varieties. Mathematics Subject Classification (2000). 14M15, 20G05.

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial. ᮊ