Tensors: Asymptotic Geometry and Developments 2016-2018
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-5136-3 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-5136-3 (ISBN)
Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. Numerous open problems appropriate for graduate students and post-docs are included throughout.
Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 Christandl-Vrana-Zuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication.
Numerous open problems appropriate for graduate students and post-docs are included throughout.
Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 Christandl-Vrana-Zuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication.
Numerous open problems appropriate for graduate students and post-docs are included throughout.
J.M. Landsberg, Texas A&M University, College Station, TX.
Basics: Motivation, first definitions and properties
Tensors via linear algebra: Rank and border rank
Tensor networks
The asymptotic geometry of tensors: Detour into probability and information theory
Strassen's laser method and spectral theory
Quantum mechanics for quantum information theory
Quantum information theory and the asymptotic geometry of tensors
Moment maps and moment polytopes
Hints and answers to selected exercises
Bibliography
Index.
| Erscheinungsdatum | 01.07.2019 |
|---|---|
| Reihe/Serie | CBMS Regional Conference Series in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 296 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-5136-0 / 1470451360 |
| ISBN-13 | 978-1-4704-5136-3 / 9781470451363 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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