Preface

Tensors are ubiquitous in the sciences. One reason for their ubiquity is

that they provide a useful way to organize data. Geometry is a powerful

tool for extracting information from data sets, and a beautiful subject in its

own right. This book has three intended uses: as a classroom textbook, a

reference work for researchers, and a research manuscript.

0.1. Usage

Classroom uses. Here are several possible courses one could give from this

text:

(1) The first part of this text is suitable for an advanced course in

multilinear algebra—it provides a solid foundation for the study of

tensors and contains numerous applications, exercises, and exam-

ples. Such a course would cover Chapters 1–3 and parts of Chapters

4–6.

(2) For a graduate course on the geometry of tensors not assuming

algebraic geometry, one can cover Chapters 1, 2, and 4–8 skipping

§§2.9–12, 4.6, 5.7, 6.7 (except Pieri), 7.6 and 8.6–8.

(3) For a graduate course on the geometry of tensors assuming alge-

braic geometry and with more emphasis on theory, one can follow

the above outline only skimming Chapters 2 and 4 (but perhaps

add §2.12) and add selected later topics.

(4) I have also given a one-semester class on the complexity of ma-

trix multiplication using selected material from earlier chapters and

then focusing on Chapter 11.

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