Parametrization, Embedding and Extension Problems in Metric Spaces

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Geometric function theory is a field of mathematics that was developed starting in the 1920s in order to study analytic functions from a geometric point of view, and was later developed to what is known today as analysis of metric spaces. The advantage of a geometric approach, is that first order differential calculus and geometric measure theory can be extended from the classical Euclidean or Riemannian settings to the realm of spaces without a priori smooth structure (such as fractal spaces). Results and techniques in geometric function theory have recently found important applications in geometric group theory, structure of manifolds and analysis on fractals. Furthermore, besides their mathematical importance, physical applications of these theories include reconstruction theory, study of thin films, control theory, graphic imaging and analysis of large data sets.This project features new approaches to three long-standing problems in the realm of geometric function theory that bring together several fields in analysis and geometry including geometric topology, sub-Riemannian geometry, PL geometry and geometric measure theory. The first problem aims at recognizing the intrinsic qualities of a metric space, from which a "nice" parametrization (e.g. quasisymmetric, Holder, bi-Lipschitz) by the Euclidean unit sphere or the Euclidean space can be recovered. The principal investigator proposes to relate forms of discrete curvature with global parametrizations in high dimensions. The second problem asks for conditions for which an embedding of a set into a Euclidean space with some desired properties (e.g. quasisymmetric, bi-Lipschitz) can be extended to the whole Euclidean space with the same properties. Finally, the third problem concerns the bi-Lipschitz embedability of big sets of a sub-Riemannian manifolds (such as the Heisenberg group) into some Euclidean space. Results in this direction will shed new light on the structure of the space and will improve our understanding of its geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.