Tensors: Geometry and Applications

J. M. Landsberg (Autor)

Buch | Softcover

439 Seiten

2021
American Mathematical Society (Verlag)
978-1-4704-7905-3 (ISBN)

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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

Part 1. Motivation from applications, multilinear algebra and elementary results
Chapter 1. Introduction
Chapter 2. Multilinear algebra
Chapter 3. Elementary results on rank and border rank
Part 2. Geometry and representation theory
Chapter 4. Algebraic geometry for spaces of tensors
Chapter 5. Secant varieties
Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors
Chapter 7. Tests for border rank: Equations for secant varieties
Chapter 8. Additional varieties useful for spaces of tensors
Chapter 9. Rank
Chapter 10. Normal forms for small tensors
Part 3. Applications
Chapter 11. The complexity of matrix multiplication
Chapter 12. Tensor decomposition
Chapter 13. $/mathbf {P}$ v. $/mathbf {NP}$
Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics
Part 4. Advanced topics
Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem
Chapter 16. Representation theory
Chapter 17. Weyman's method
Hints and answers to selected exercises

Erscheinungsdatum	08.07.2025
Reihe/Serie	Graduate Studies in Mathematics
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-7905-2 / 1470479052
ISBN-13	978-1-4704-7905-3 / 9781470479053
Zustand	Neuware