Metric Algebraic Geometry

This book focuses on modern applications of metric algebraic geometry. Building on classical foundations, the field embarks towards a new paradigm that combines concepts from algebraic geometry and differential geometry, with the goal of developing practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances. These require tools from different areas in geometry, algebra and analysis. Starting with a short dive into 19th-century geometry of plane curves, the book addresses a number of problems in the sciences, leading to solving polynomial equations over the real numbers. The solution sets are real algebraic varieties. Other metric problems discussed arise in optimization and statistics and involve, for instance, minimizing Wasserstein distances in machine learning, maximum likelihood estimation, or minimizing the Euclidean distance from a variety to a given data point. This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level, or for self-study. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra, geometry and numerics. Course work in statistics and computer science, as well as experience with mathematical software, are helpful. This is an open access book.