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Rectifiability of Measures in Euclidean and Metric Spaces

欧几里得和度量空间中测度的可修正性

基本信息

批准号：
1763973
负责人：
Raanan Schul
金额：
$ 18万
依托单位：
SUNY at Stony Brook
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1763973&HistoricalAwards=false
关键词：
Rectifiability Measures Euclidean Metric Spaces

项目摘要

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.

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，我们希望得到什么样的结构？有多少数据我们可以期望以有用的方式组织起来？如果我们愿意在这个过程中“丢失”一些信息，答案会改变吗？我们会知道数据丢失的数量吗？最后但并非最不重要的是，我们能否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，人们被给定一个表示为度量空间的子集的大数据集，例如对于大维度d的R^d，并试图“忠实地”表示这个数据集的“大”部分作为R^k的子集，对于维度k比d“小”得多。“忠实地”在这里意味着人们仍然可以对数据部分的图像执行相同的数据挖掘任务。这项任务迄今为止已经得到了计算机科学家和应用数学家的广泛关注，他们使用了各种各样的方法。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。几何测度理论和几何函数理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，通常给定的数据集有一些额外的几何结构，例如小Hausdorff维数（离散模拟），或接近低维流形的并集。这允许使用调和分析和几何测量理论。该项目旨在研究由此激发的数学问题。要讨论的基本问题可以表述为“什么时候度量测度空间的一部分由”标准“片的Lipschitz像组成，我们如何找到这些片？或者“什么时候点的集合最好用低维的方式来描述？“".该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。