Flat Forms, Bi-Lipschitz Parametrizations, and Calculus on Singular Spaces

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Singular (or nonsmooth) objects arise everywhere in mathematics. To mention just a few instances, they turn up in the following situations: as limits of smooth objects (spaces such as Riemannian manifolds, or smooth functions); as asymptotic "spheres at infinity" of finitely generated groups, objects that reflect the behavior of the groups at large scales; or in the fine-scale structure of sets, even in concrete and practical circumstances (irregular crystals, for instance, or other materials). The principal investigator studies questions that relate to singular objects and their geometry, and he does analysis on such objects. Specifically, he seeks to understand the extent to which the concepts of classical differential (first-order) analysis can be introduced into such spaces. For example, one would like to have a well-defined Sobolev space of weakly differentiable functions on certain singular spaces. It is also important to understand which potentially very singular spaces can be parametrized by "nice" spaces (say, by Euclidean spaces) via transformations that distort the basic metric structure only within fixed bounds. The principal investigator and his students are developing new tools for approaching this type of problem. Finally, the question of parametrization by a Euclidean space can be replaced with the requirement of embedability in some finite-dimensional Euclidean space. The methods that emerge from the project should clarify this problem as well.The proposed research relates to applications in two ways. First, singularities (or impurities) occur everywhere in nature, from the local microstructure of materials to the large-scale features of the universe. Understanding and dealing with such singularities is one of the central objectives of modern mathematics and science. The principal investigator has made contributions to the solution of this problem in cases where the singularities can be analyzed and then transformed, with minimal cost, to better behaved models. Second, although not directly related to the project, there are potential applications of the research to theoretical computer science, where large and complex data sets need to be transformed and stored in simpler form.