Metric Algebraic Geometry: Going Global

This Semester Program will explore the emerging research field of Metric Algebraic Geometry, which is about the geometry induced by metric and probabilistic structures on sets defined by polynomial equations. The philosophy of this program is that regarding spaces and models as algebraic varieties opens up powerful – notably, global - approaches for problems involving distances and/or approximation. Instead of viewing spaces through local charts or via simplicial pieces, Metric Algebraic Geometry proposes finding global descriptions. This brings new computational tools to the table to establish strong theoretical properties over the geometric domain. Target applications include the nonlinear function spaces of neural networks, reduced-order models for large datasets, and configuration spaces of cameras or images in computer vision. The program will develop this area both within pure mathematics and in its applications. Activities will encourage experts in theoretical fields (such as algebraic geometry, differential geometry, and integral geometry) to commingle with experts in application-driven domains (like in machine learning, computer vision, optimization, nonlinear inverse problems and probabilistic modeling) and to all mentor the next generation. The results will set the research agenda for Metric Algebraic Geometry for years to come.