Metric Algebraic Geometry
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Beschreibung
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
This is an open access book.
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
This is an openaccess book.
brings algebraic and differential geometry together in a computational setting provides a modern view on this connection, motivated by data science and AI focuses on concrete examples from a wide variety of applications This book is open access, which means that you have free and unlimited access
Autor*in
Paul Breiding
Themen in »Metric Algebraic Geometry«
Algebraic Variety Data Science Differential Geometry Euclidean Distance Integrals Maximum Likelihood Numerical Methods Polynomial System Tensors Curvature Polynomial Optimization Open Access
Stimmen zu »Metric Algebraic Geometry«
“This interdisciplinary work is a relatively short, well-written, and beautifully illustrated book laying key foundations for the emerging field of metric algebraic geometry. … The book should be widely accessible since the key prerequisite is a solid foundation in undergraduate mathematics, especially algebra and geometry.” (Scott McCallum, Mathematical Reviews, October, 2025)
“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 Zaldiva, MAA Reviews, May 2, 2024)
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