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Metric Algebraic Geometry

Paul Breiding, Kathlén Kohn, Bernd Sturmfels
Publication Date: 
Number of Pages: 
[Reviewed by
Felipe Zaldivar
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Metric algebraic geometry aims to join methods and ideas of (metric) differential geometry with its (real) algebraic counterpart, with a common goal a variety of practical applications in optimization, modeling and analyzing statistical data, machine learning and computer vision. An ambitious goal indeed. The motivating observation is that many problems in those applications can be approached using polynomial equations in several variables and with coefficients in the field of real numbers. The metric properties (volumes, areas, distances, angles) of the real algebraic varieties defined by the corresponding polynomial equations are often the needed tools to solve or better understand de associated application. These properties and methods are somehow well understood in the case of (linear) optimization problems and part of this new proposal is to put in the same footing the problems that lead to polynomial equations of higher degree. Several recent techniques aim to achieve similar goals: tropical algebraic geometry, algebraic statistics or topological data analysis, together with the ever ubiquitous Gröbner bases as a calculation method. To these arsenal, the authors propose adding now tools from differential geometry. A sample of these techniques include Wassernstein metrics from polyhedral norms with applications to optimal transport or tensors and their ranks with applications to 3-dimensional reconstruction in computer vision. There are more of these techniques and applications and many readers will certainly find one close to their interests.  The fifteen chapters of the book are well motivated and within reach of a motivated reader with a moderate background in geometry and unrelated fields of application, for example, in probability theory or statistical models.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is