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Metric Algebraic Geometry
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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.
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Keywords
- Algebraic Geometry
- Algebraic Variety
- Calculus & mathematical analysis
- Computing & information technology
- curvature
- Data science
- Databases
- Differential & Riemannian geometry
- Differential Geometry
- Euclidean distance
- Geometry
- Integrals
- Mathematics
- Mathematics & science
- maximum likelihood
- Numerical analysis
- Numerical methods
- Polynomial Optimization
- Polynomial System
- tensors
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMW Algebraic geometry
- thema EDItEUR::U Computing and Information Technology::UN Databases
Links
DOI: 10.1007/978-3-031-51462-3Editions