Rectifiability of Measures in Euclidean and Metric Spaces

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One aspect of modern technology is that it is easy to collect data. A very challenging task is to sift through a large collection of data in order to find meaningful information. One would like to organize the data, or at least part of it, in such a way that it is easy to use. Imagine the data as being all images on the internet, and the "organization" that you seek is being able to map out which images are those of a specific person of your choosing, sub-ordered according to the activities in which he or she is engaged. Organizing large amounts of information in a useful way, or sorting through it and finding pieces you care about, are tasks that can be transformed, or related to, mathematical questions. This proposal attempts to address some of these questions. Basic questions are mathematical analogues of the following: What kind of structure can we hope to get after organizing the data? How much of the data can we expect to organize in a useful way? Do answers change if we are willing to "lose" some information in the process? Will we know the amount data lost? And, last but not least, can we, in a practical way, access the organized data or a significant part of it?In many applications one is given a large data set represented as a subset of a metric space, such as R^d for large dimension d, and seeks to `faithfully' represent a `large' portion of this data set as a subset of R^k for dimension k much `smaller' than d. `Faithfully' here, means that one can still perform the same data mining tasks on the image of the data portion. This task has thus far yielded much attention from computer scientists and applied mathematicians using a wide range of approaches. The framework of dimensionality reduction also includes data compression and data approximation. These have applications in many areas of science. Geometric Measure Theory and Geometric Function Theory are tools whose use in this matter has not been fully exploited. A key point is that often the given data set has some additional geometric structure, for example small Hausdorff dimension (a discrete analogue), or being close to a union of low dimensional manifold. This allows one to use harmonic analysis and geometric measure theory. The project aims at studying mathematical questions motivated by this. Basic questions to be discusses can be phrased as "When is part of a metric measure space composed of Lipschitz images of `standard' pieces and how do we find these pieces?" or "When is a collection of points best described in a low-dimensional way?". The tools to be used come from a combination of Harmonic Analysis and Geometric measure theory, which is usually referred to as quantitative rectifiability.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.