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oapen-20.500.12657-883602024-03-28T14:02:54Z Metric Algebraic Geometry Breiding, Paul Kohn, Kathlén Sturmfels, Bernd Algebraic Variety Data Science Differential Geometry Euclidean Distance Integrals Maximum Likelihood Numerical Methods Polynomial System Tensors Curvature Polynomial Optimization thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMW Algebraic geometry thema EDItEUR::P Mathematics and Science::PB Mathematics::PBM Geometry::PBMP Differential and Riemannian geometry thema EDItEUR::U Computing and Information Technology::UN Databases thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis 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. 2024-03-13T11:11:13Z 2024-03-13T11:11:13Z 2024 book ONIX_20240313_9783031514623_48 9783031514623 9783031514616 https://library.oapen.org/handle/20.500.12657/88360 eng Oberwolfach Seminars application/pdf n/a 978-3-031-51462-3.pdf https://link.springer.com/978-3-031-51462-3 Springer Nature Birkhäuser 10.1007/978-3-031-51462-3 10.1007/978-3-031-51462-3 6c6992af-b843-4f46-859c-f6e9998e40d5 8e7742bc-c35e-4523-8581-875fb10bfc5e d880bcc0-ecff-409c-9896-c00d438cc124 9783031514623 9783031514616 Birkhäuser 53 215 Cham [...] [...] Max-Planck-Institut für Mathematik in den Naturwissenschaften Max Planck Institute for Mathematics in the Sciences open access
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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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