**CBMS Regional Conference Series in Mathematics**

Volume: 132;
2019;
144 pp;
Softcover

MSC: Primary 68; 14; 15; 81;

**Print ISBN: 978-1-4704-5136-3
Product Code: CBMS/132**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-5291-9
Product Code: CBMS/132.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

# Tensors: Asymptotic Geometry and Developments 2016–2018

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*J.M. Landsberg*

A co-publication of the AMS and CBMS

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.

#### Readership

Undergraduate and graduate students and researchers interested in tensors, mathematical aspects of solid state physics, and quantum information theory.

#### Table of Contents

# Table of Contents

## Tensors: Asymptotic Geometry and Developments 2016-2018

- Cover Cover11
- Title page iii4
- Preface vii8
- Part 1 . Basics 112
- Part 2 . Tensors via linear algebra 1930
- Part 3 . The asymptotic geometry of tensors 4960
- Chapter 4. Detour into probability and information theory 5162
- Chapter 5. Strassen’s laser method and spectral theory 5970
- Chapter 6. Quantum mechanics for quantum information theory 7990
- Chapter 7. Quantum information theory and the asymptotic geometry of tensors 95106
- Chapter 8. Moment maps and moment polytopes 115126
- Hints and Answers to Selected Exercises 129140
- Bibliography 133144
- Index 141152

- Back Cover Back Cover1158