Tensors: Asymptotic Geometry and Developments 2016-2018

Just as with their 2-dimensional versions, tensors can be used to organize data in multidimensional arrays. Multilinear algebra provides the necessary background, but it is the geometry of tensors the new powerful tool that enlightens the structure of the data sets and helps to extract the information gathered in these sets, once this information is organized in tensors.  Applications of the new tools provided by representation theory and algebraic geometry range from algebraic statistics to signal analysis,  quantum information theory, condensed matter physics and complexity theory. 

 

The book under review expands and updates some topics of the previous books by the author, Tensors; Geometry and Applications and Geometry and Complexity Theory.   The book collects various recent developments on some specific topics on rank additivity and symmetric rank, the asymptotics of matrix multiplication, moment polytopes, quantum information theory and tensor networks in solid-state physics, all of which are current and active areas of research.

 

A terse first chapter summarizes the basic facts on tensors,  rank and border rank, algebraic varieties, linear group actions and their Lie algebra counterparts. Chapter six gives a quick utilitarian overview of quantum information theory phrasing everything in terms of linear algebra, from probabilistic (classical or quantum) computations to quantum channels and entanglement. These two chapters make the book somewhat self-contained by providing the background needed for the remaining chapters.

 

Graduate students and researchers with an interest on applications of algebraic geometry and representation theory will find this book timely and filled with many exciting new results, nicely updating the expositions quoted above.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx