On the Whitney extension problem for near isometries and beyond
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This paper is an exposition of work of the author et al. detailing fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic-differential geometry, analysis on manifolds, complex-harmonic analysis, data science, partial differential equations, optimization and probability. A significant portion of the work is based on joint research with Charles Fefferman in the papers [39, 40, 41, 42]. The topics of this work include (a) The space of maps of bounded mean oscillation (BMO) in ℝ^D, D≥ 2. (b) The labeled and unlabeled near alignment and Procrustes problem for point sets with certain geometries and for not too thin compact sets both in ℝ^D, D≥ 2. (c) The Whitney near isometry extension problem for point sets with certain geometries and for not too thin compact sets both in ℝ^D, D≥ 2. (d) Partitions and clustering of compact sets and point sets with certain geometries in ℝ^D, D≥ 2 and analysis on certain manifolds in ℝ^D, D≥ 2. Many open problems for future research are given.
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