Geometric Measure Theory and Geometric Function Theory

现代技术的一个方面是很容易收集数据。 一个非常具有挑战性的任务是筛选大量的数据，以便找到有意义的信息。 人们希望以易于使用的方式组织数据，或至少组织其中的一部分。 想象一下，这些数据是互联网上的所有图像，你所寻求的“组织”能够映射出哪些图像是你选择的特定人的图像，根据他或她所从事的活动进行细分。 以一种有用的方式组织大量的信息，或者对它们进行分类并找到你关心的部分，这些任务可以转化为数学问题，或者与数学问题相关。 本提案试图解决其中一些问题。 基本问题是以下数学类比：在组织数据后，我希望得到什么样的结构？ 我可以期望以有用的方式组织多少数据？如果我愿意在这个过程中“丢失”一些信息，答案会改变吗？我会知道数据丢失的数量吗？最后但并非最不重要的是，我是否可以以实际的方式访问组织好的数据或其中的重要部分？在许多应用中，人们被赋予一个表示为度量空间（如高维欧几里得空间）的子集的大数据集，并且人们试图将该数据集的“大”部分“忠实地”表示为低维欧几里得空间的子集。“忠实地”在此上下文中意味着人们仍然可以在数据部分的图像上执行与在原始数据集上相同的数据挖掘任务。 这项任务迄今已受到计算机科学家和应用数学家的广泛关注，使用各种各样的方法。降维的框架还包括数据压缩和数据逼近。这些在许多科学领域都有应用。 几何测度论和几何函数论是其在这一问题上的使用尚未得到充分利用的工具。一个关键点是，通常给定的数据集具有一些额外的几何结构，例如，具有小的Hausdorff维数（离散模拟）或接近于低维流形的并集。这允许使用调和分析和几何测量理论来组织数据。 这个项目将研究由这一观察激发的数学问题。该项目将试图回答的两个基本问题可以表述如下：什么时候度量测度空间的一部分由“标准”片段的Lipschitz图像组成，以及如何找到这些片段？什么时候点的集合最好描述为一维？

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, subordered 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 I hope to get after organizing the data? How much of my data can I expect to organize in a useful way? Do answers change if I am willing to "lose" some information in the process? Will I know the amount data lost? And, last but not least, can I, 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 a high-dimensional Euclidean space, and one seeks to "faithfully" represent a "large" portion of this data set as a subset of a low-dimensional Euclidean space. "Faithfully" means in this context that one can still perform the same data mining tasks on the image of the data portion that one could on the original data set. This task has thus far received 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, has small Hausdorff dimension (a discrete analogue) or is close to being a union of low-dimensional manifolds. This allows one to use harmonic analysis and geometric measure theory to organize the data. This project will study mathematical questions motivated by this observation. Two basic questions the project will attempt to answer can be phrased as follows: When is part of a metric measure space composed of Lipschitz images of "standard" pieces and how does one find these pieces? When is a collection of points best described as one-dimensional?