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:
- 金额:$ 18万
- 依托单位:
- 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:
- 关键词:
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。
项目成果
期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Lower bounds on mapping content and quantitative factorization through trees
通过树映射内容和定量分解的下限
- DOI:10.1112/jlms.12595
- 发表时间:2022
- 期刊:
- 影响因子:0
- 作者:David, Guy C.;Schul, Raanan
- 通讯作者:Schul, Raanan
Iterating the Big‐Pieces operator and larger sets
迭代 Big-Pieces 运算符和更大的集合
- DOI:10.1112/blms.12683
- 发表时间:2022
- 期刊:
- 影响因子:0.9
- 作者:Krandel, Jared;Schul, Raanan
- 通讯作者:Schul, Raanan
A sharp necessary condition for rectifiable curves in metric spaces
度量空间中可矫正曲线的尖锐必要条件
- DOI:10.4171/rmi/1216
- 发表时间:2021
- 期刊:
- 影响因子:0
- 作者:David, Guy;Schul, Raanan
- 通讯作者:Schul, Raanan
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Raanan Schul其他文献
Multiscale Analysis of 1-rectifiable Measures II: Characterizations
1-可纠正措施的多尺度分析 II:特征
- DOI:
- 发表时间:
2016 - 期刊:
- 影响因子:0
- 作者:
Matthew Badger;Raanan Schul - 通讯作者:
Raanan Schul
Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian
通过热核和拉普拉斯本征函数的通用局部参数化
- DOI:
10.5186/aasfm.2010.3508 - 发表时间:
2007 - 期刊:
- 影响因子:0
- 作者:
Peter W. Jones;M. Maggioni;Raanan Schul - 通讯作者:
Raanan Schul
Two sufficient conditions for rectifiable measures
纠正措施的两个充分条件
- DOI:
- 发表时间:
2014 - 期刊:
- 影响因子:0
- 作者:
Matthew Badger;Raanan Schul - 通讯作者:
Raanan Schul
Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
Bi-Lipschitz 将 Lipschitz 函数分解为度量空间
- DOI:
- 发表时间:
2007 - 期刊:
- 影响因子:0
- 作者:
Raanan Schul - 通讯作者:
Raanan Schul
Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
Hard Sard:Lipschitz 映射的定量隐式函数和可拓定理
- DOI:
10.1007/s00039-012-0189-0 - 发表时间:
2011 - 期刊:
- 影响因子:2.2
- 作者:
Jonas Azzam;Raanan Schul - 通讯作者:
Raanan Schul
Raanan Schul的其他文献
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{{ truncateString('Raanan Schul', 18)}}的其他基金
Geometry of Sets and Measures in Euclidean and Non-Euclidean Spaces
欧几里得和非欧空间中的集合和测度的几何
- 批准号:
2154613 - 财政年份:2022
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
Conference on Analysis, Dynamics, Geometry, and Probability
分析、动力学、几何和概率会议
- 批准号:
1954590 - 财政年份:2020
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
Conference in Geometry, Analysis, and Probability
几何、分析和概率会议
- 批准号:
1700209 - 财政年份:2017
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
Geometric Measure Theory and Geometric Function Theory
几何测度论和几何函数论
- 批准号:
1361473 - 财政年份:2014
- 资助金额:
$ 18万 - 项目类别:
Continuing Grant
Harmonic Analysis, Geometric Measure Theory and Applications
调和分析、几何测量理论及应用
- 批准号:
1100008 - 财政年份:2011
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
Harmonic Analysis and Faithful Data Representations. Multiscale Analysis and Diffusion Processes
谐波分析和忠实的数据表示。
- 批准号:
0965766 - 财政年份:2009
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
Harmonic Analysis and Faithful Data Representations. Multiscale Analysis and Diffusion Processes
谐波分析和忠实的数据表示。
- 批准号:
0800837 - 财政年份:2008
- 资助金额:
$ 18万 - 项目类别:
Standard Grant
PostDoctoral Research Fellowship in the Mathematical Sciences
数学科学博士后研究奖学金
- 批准号:
0502747 - 财政年份:2005
- 资助金额:
$ 18万 - 项目类别:
Fellowship
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