Analysis and geometry of metric measure spaces

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Geometry and analysis have become increasingly important to real world applications in the last few decades. We are now capable of gathering immense amounts of data, and so it is a central concern to develop quick and efficient ways of analyzing this data. In many instances, one can endow the points of a data set with a notion of distance and weight, thereby giving us a geometric representation of the data. One can then try to take advantage of this geometry and leverage this information to speed up computations on the data set. This project seeks to develop new techniques and theorems for understanding the geometric and analytic structure of spaces with some notion of distance and (possibly) weight. Some basic questions the project will address are the following: Can I represent my space in some other space (e.g. Euclidean space) so that the geometry is preserved with high fidelity? Does my space exhibit low dimensional behavior despite living in a possibly high (or infinite) dimensional space? Is there a calculus available on my space that I can use to analyze functions on the space?This project will study analytic and geometric properties of certain classes of metric measure spaces. In the first part of the project, the PI will study problems about metric embeddings including whether doubling subsets of Hilbert space embed into Euclidean spaces and developing biLipschitz metric invariants to measure uniform smoothness. The second part of the project will focus on detecting rectifiable behavior in metric measure spaces. Rectifiable spaces admit parameterizations by simple model spaces (usually Euclidean space) and so can exhibit low dimensional behavior. The PI will study Carnot notions of rectifiability and also link these geometric properties to boundedness of singular integrals. Finally, the PI will continue his study into the nature of Lipschitz differentiability on metric measure spaces as introduced by Cheeger. The project will determine whether such differentiation differs between real valued and RNP Banach space valued functions. If these notions turn out to be different, the PI will then seek to discover the geometric mechanisms for the former (the PI has already been geometrically characterized the latter). The PI will also seek to broaden the differentiability theory to non-Banach space target.