Geometric Measure Theory and Geometric Function Theory

现代技术的一个方面是很容易收集数据。一项非常具有挑战性的任务是筛选大量的数据，以找到有意义的信息。人们希望以一种易于使用的方式来组织数据，或者至少组织其中的一部分。想象一下，这些数据都是互联网上的所有图片，你要寻找的“组织”能够标明哪些图片是你选择的特定人的图片，并根据他或她所从事的活动进行细分排序。以有用的方式组织大量信息，或对其进行分类并找到您关心的部分，这些都是可以转换的任务，或者与数学问题相关的任务。这项提案试图解决其中一些问题。基本问题是对以下问题的数学类比：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我的数据中的多少？如果我愿意在这个过程中“丢失”一些信息，答案会改变吗？我会知道丢失的数据量吗？最后但并非最不重要的一点是，我能以一种实用的方式访问组织的数据或其重要部分吗？在许多应用中，人们被给予表示为度量空间的子集的大数据集，例如高维欧几里德空间，并且人们寻求将该数据集的一大部分忠实地表示为低维欧几里德空间的子集。在这种情况下，“忠实地”意味着人们仍然可以对数据部分的图像执行与对原始数据集相同的数据挖掘任务。到目前为止，这项任务受到了计算机科学家和应用数学家的极大关注，他们使用了各种方法。降维的框架还包括数据压缩和数据逼近。它们在许多科学领域都有应用。几何测度论和几何函数论是尚未被充分利用的工具。一个关键点是，给定的数据集通常具有一些额外的几何结构，例如，具有较小的Hausdorff维度(离散模拟)或接近于低维流形的并集。这使得人们可以使用调和分析和几何测量理论来组织数据。这个项目将研究受此观察启发的数学问题。该项目将试图回答的两个基本问题可以表述如下：公制度量空间的一部分何时由利普希茨图像组成，以及如何找到这些图像？什么时候点的集合最适合描述为一维？

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?