Harmonic Analysis, Geometric Measure Theory and Applications

从数学上讲，该项目涉及几何测量理论（GMT）和谐波分析。更具体地说，我们在这两个领域的界面上研究问题，并使用谐波分析技术来研究GMT中的问题。我们研究这样的问题：“从给定度量空间的显著部分到欧几里得空间找到一个biLipschitz映射”，“刻画欧几里得平面的biLipschitz像”，或者“度量空间何时由标准块的biLipschitz像构成，我们如何找到这些块？”在谐波分析方面，这与这样的问题有关：“一个函数什么时候可以分解成一系列好的函数的和，以及如何构造这个分解？”最后一个问题是Littlewood-Paley和小波分析中的标准问题，将这些丰富的理论方法转移到GMT设置中产生了一个重要的工具箱。这种类型的GMT研究与应用中出现的许多问题密切相关。在许多应用中，给定一个大数据集，表示为度量空间的子集，例如高维欧几里得空间，并且试图忠实地将该数据集的很大一部分表示为低维欧几里得空间的子集。忠实地，在这里意味着仍然可以对数据部分的图像执行相同的数据挖掘任务。正是由于这种与数据挖掘的联系，上述任务迄今为止已经引起了计算机科学家和应用数学家的广泛关注，并使用了广泛的方法。降维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；例如文档分析，人脸识别，聚类，机器学习，非线性图像去噪，分割和处理。该项目旨在更好地理解给定空间中点集合的几何形状。在许多应用中，我们将给定的数据视为度量空间中的点，我们希望将它们映射到我们更好地理解的低维空间（例如低维欧几里德空间）。该数据挖掘任务通常称为降维，该框架包括数据压缩和数据近似。这些在许多科学领域都有应用；例如文档分析，人脸识别，聚类，机器学习，非线性图像去噪，分割和处理。关键的一点是，数据通常具有良好的几何属性，并且比随机的点集具有更多的结构。人们可以研究这些几何结构并使用它们来执行数据挖掘任务。在这些几何结构的研究背后有丰富的数学理论，这就是我们所发展的。

Mathematically speaking, the project deals with Geometric Measure Theory (GMT) and Harmonic Analysis. More specifically, we study questions on the interface of both these fields, and use harmonic analysis techniques to study questions in GMT. We study problems of the form "Find a biLipschitz map from a significant part of a given metric space to Euclidean space", "Characterize biLipschitz images of the Euclidean plane" or "When is a metric space built from biLipschitz images of standard pieces and how do we find these pieces?" On the harmonic analysis side, this is related to questions of the form "When is a function decomposable into a sum of nice functions, and how do you construct this decomposition?" This last question is a standard one in Littlewood-Paley and wavelet analysis, and transferring the methods from these rich theories into the setting of GMT yields a significant toolbox. This type of study of GMT is very related to many questions that arise in applications. 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 here, means that one can still perform the same data mining tasks on the image of the data portion. It is because of this connection to data mining that the above 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; for examples document analysis, face recognition, clustering, machine learning nonlinear image denoising, segmentation and processing.The project is geared towards a better understanding of the geometry of collections of points in a given space. In many applications one is given data which we think of as points in a metric space, and we want to map them to a low dimensional space which we understand better (such as a low dimensional Euclidean space). This data-mining task is typically called dimensionality reduction, and this framework includes data compression and data approximation. These have applications in many areas of science; for examples document analysis, face recognition, clustering, machine learning nonlinear image denoising, segmentation and processing. A key point is that quite often the data enjoys nice geometric properties and has more structure then a random set of points would have. One can study these geometric structures and use them to perform data mining tasks. There is a rich mathematical theory behind the study of such geometric structures, and this is what we develop.