Geometry of Sets and Measures in Euclidean and Non-Euclidean Spaces

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The modern world is awash in data. The task of organizing large amounts of data in useful and ordered ways can be formulated in mathematical terms. This project investigates mathematical analogs of questions such as the following: How much data can we expect to organize in a useful way? What type of geometric structures arise in the process of such organization? Do these answers change if we allow ourselves to disregard a certain amount of information, and can the impact of such a choice be quantified? Finally, are there practical algorithms to implement such data organization? In geometric language, data naturally resides in a high dimensional space or a space where the notion of distance is quite different from the Euclidean one. This project aims at transferring well studied and efficient tools for analysis from low-dimensional Euclidean spaces to higher-dimensional and more general settings, allowing high-dimensional data to be "visualized" in a lower-dimensional, structured environment. The project will involve the training and mentoring of graduate students and postdocs and aims to develop tools which can lead to engagement between pure mathematicians and the data science community. In many applications one is given a large data set, represented as a subset of a high-dimensional space, and one seeks to faithfully represent a large portion of this data set in a space of substantially lower dimension. "Faithfully" here means that essential geometric features are either preserved or mildly distorted. The Lipschitz condition for a geometric transformation quantifies the distortion of distances between data points. To date, the preceding task has received attention from computer scientists and applied mathematicians using a range of approaches. This project investigates mathematical approaches rooted in analysis and geometry. A key point is that often the given data has additional geometric structure, for example, it may have small Hausdorff dimension or be close to a union of low dimensional manifolds. Such added structure allows for the use of tools from harmonic analysis and geometric measure theory, especially, the theory of rectifiability. A quantitative version of this theory, known as uniform rectifiability, will be explored in novel metric settings. Other topics to be considered include quantitative improvements of low rank factorization theorems, Lipschitz decompositions of metric measure spaces, low-distortion factorization of bi-Lipschitz mappings, and Lipschitz parameterizations of high-dimensional spaces with parameterizing dimension greater than one. 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.