Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

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viii Contents §8.6. The Chow variety of zero cycles and its equations 221 §8.7. The Fano variety of lines on a variety 225 Chapter 9. Rank 229 §9.1. Remarks on rank for arbitrary varieties 229 §9.2. Bounds on symmetric rank 231 §9.3. Examples of classes of polynomials and their ranks 235 Chapter 10. Normal forms for small tensors 243 §10.1. Vector spaces with a finite number of orbits 244 §10.2. Vector spaces where the orbits can be explicitly parametrized246 §10.3. Points in C 2 ⊗C b ⊗C c 249 §10.4. Ranks and border ranks of elements of S 3 C 3 255 §10.5. Tensors in C 3 ⊗C 3 ⊗C 3 257 §10.6. Normal forms for C 2 ⊗S 2 W 258 §10.7. Exercises on normal forms for general points on small secant varieties 259 §10.8. Limits of secant planes 259 §10.9. Limits for Veronese varieties 261 §10.10. Ranks and normal forms in σ3(Seg(PA1⊗ · · · ⊗ PAn)) 264 Part 3. Applications Chapter 11. The complexity of matrix multiplication 271 §11.1. “Real world” issues 272 §11.2. Failure of the border rank version of Strassen’s conjecture 273 §11.3. Finite group approach to upper bounds 277 §11.4. R(M3,3,3) ≤ 23 279 §11.5. Bläser’s 5 2-Theorem 279 §11.6. The Brockett-Dobkin Theorem 281 §11.7. Multiplicative complexity 283 Chapter 12. Tensor decomposition 285 §12.1. Cumulants 286 §12.2. Blind deconvolution of DS-CMDA signals 289 §12.3. Uniqueness results coming from algebraic geometry 295 §12.4. Exact decomposition algorithms 297 §12.5. Kruskal’s Theorem and its proof 301
Contents ix Chapter 13. P v. NP 307 §13.1. Introduction to complexity 308 §13.2. Polynomials arising in complexity theory, graph theory, and statistics 311 §13.3. Definitions of VP, VNP and other algebraic complexity classes 313 §13.4. Complexity of perm n and detn 318 §13.5. Immanants and their symmetries 325 §13.6. Geometric Complexity Theory approach to VPws v. VNP 329 §13.7. Other complexity classes via polynomials 336 §13.8. Vectors of minors and homogeneous varieties 337 §13.9. Holographic algorithms and spinors 344 Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics 353 §14.1. Tensor network states 353 §14.2. Algebraic Statistics and phylogenetics 359 Part 4. Advanced topics Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem 369 §15.1. The semi-classical cases 370 §15.2. The Alexander-Hirschowitz idea for dealing with the remaining cases 373 Chapter 16. Representation theory 377 §16.1. Basic definitions 377 §16.2. Casimir eigenvalues and Kostant’s theorem 381 §16.3. Cohomology of homogeneous vector bundles 385 §16.4. Equations and Inheritance in a more general context 389 Chapter 17. Weyman’s method 391 §17.1. Ideals and coordinate rings of projective varieties 392 §17.2. Koszul sequences 393 §17.3. The Kempf-Weyman method 396 §17.4. Subspace varieties 400 Hints and Answers to Selected Exercises 405 Bibliography 411
Page 1: Tensors: Geometry and Applications
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Page 10 and 11: xii Preface (5) Similarly I have us
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