Characterization of Trace Spaces and Differential Structures on Subsets of Euclidean Space

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This research project is focused on applications of extension theory to problems in geometry, partial differential equations, computer science, and data processing. In the machine learning paradigm, one typically observes a collection of data that represents the measurements of a physical process. The data may be represented in terms of a large number of variables. To learn the physical structures inherent to the data, one looks for a small number of latent variables that describe a low-dimensional manifold which contains or "passes close to" the data points. Whereas classical statistical regression looks for a linear relationship in the data, manifold learning allows for possibly non-linear descriptions. The principal investigator plans to develop practical manifold learning algorithms using techniques from extension theory. Consider for instance the problem of interpolating a set of data by a smooth, convex hypersurface. Whereas all currently available solutions to this problem require that the data points be sampled uniformly from the entire surface, the principal investigator proposes an approach which would work even when there are no sampled data on large pieces of the surface. The proposed research will introduce new connections between the fields of harmonic analysis and machine learning.The past few decades have witnessed the development of abstract notions of differentiation and curvature on nonsmooth spaces. To study singular solutions to mean curvature flow, for instance, it is important to develop a generalized notion of curvature. Another achievement has been Cheeger's theory of differentiation on metric measure spaces. Cheeger's spaces carry a notion of distance and volume, but surprisingly, they lack the structure of local coordinate charts usually required to define a differential calculus. One shortcoming of this approach is its inherent limitation to first order theories. That is, one can define a first order differential operator, but the notion of a second order operator, such as the Laplacian or heat operator, is meaningless at this level of abstraction. The principal investigator proposes an alternate perspective: Assume the space is embedded as a subset of a Euclidean space. One can take the differential calculus on the Euclidean space and push it forward to define a calculus on the subset. This simple technique allows one to define high-order tangent and cotangent bundles on subsets of Euclidean space. An issue with the approach is that the computation of the bundles is highly nontrivial. The first order tangent bundle can be defined by a standard blow-up argument, but the higher order "paratangent bundles" are difficult to understand. By focusing on a class of explicit examples of algebraic varieties with cuspidal singularities, the principal investigator will find new methods for computing with these abstract spaces. The principal investigator will develop a notion of divided difference quotients on algebraic and semialgebraic sets.