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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.
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I am no specialist on this subject, so I found Tensors difficult but fascinating. ...The exposition is terse, very much in the style of a graduate textbook. The reader must work through the book and become conversant with the subject. ... Most readers will enjoy the preface and chapter 1, which set out the main problems and the motivation from applied mathematics. ...A reader who knows linear and multilinear algebra and wants to know more about these questions could read Part 1 with profit. Part 2 is where the real work is done, with algebraic geometry and representation theory being the main tools. The text gets significantly denser. There is a lot of mathematics here, enough for a graduate course on this material. Part 3 returns to the applications and puts the theory to use. Part 4 is a kind of supplement that gives proofs that require more advanced techniques and discusses other advanced topics. --MAA Reviews
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