Geometric and Combinatorial Viewpoints in Complex and Harmonic Analysis

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。关于该项目的知识内容，该提案的目的是研究准共形（QC）映射，几何分析（特别是一致可求长性），傅立叶分析和几何组合学之间的相互作用。更具体地说，PI将追踪（1）K-QC图的尖锐失真示例。(2)有界拟正则（QR）映射下集合可去性的尖锐充分条件（与尖锐QC失真定理有关）(3)Cantor集的Buffon针概率（Favard长度）（4）涉及一致可求长集与调和测度的问题(5)有关傅立叶分析和几何组合学的距离集问题。所采用的统一方法是一种潜在的几何组合视觉，它通常通过多尺度分析（即在不同尺度上分析一个问题）来体现。这种方法将被应用于K-QC映射的上下文中（mappings sending an infinitesimal circle/ball to an infinitesimal ellipse/ellipsoid with eccentricity controlled by K），几何测度论（GMT，分析集合和度量-这些是长度，面积和体积的概括-），谐波分析（将信号分解为具有波动特征的基本片段）和势能理论（研究库仑势和相关主题）。 关于在更广泛的数学和科学框架内将拟议的研究置于背景中，请注意所涉及的数学对象在其他学科中有大量的应用，因此提出的问题将促进这些领域的知识，从而影响数学，科学或工程的其他领域。更具体地说，分形（几何测量理论）自然出现在电沉积和扩散限制聚集。肺的内部结构具有较高的分形维数（以捕获更多的氧气）。傅立叶分析经常应用于信号和图像处理。拟共形映射是非线性弹性问题的解，并在弦理论中得到应用。均匀可校正性出现在Mumford-Shah泛函的最小化器中（最初用于图像分割）。几何组合学用于社会科学中的公平分配和投票问题，以及生物学中的系统发育树模型。距离集在工业中用于研究数据集的维数。在人力资源开发方面，PI将继续为学生准备普特南竞赛，参加数学俱乐部，并在研究生课程中非正式地指导研究生。分形和几何组合学是促进本科生和博士后教学和培养的优秀领域。多尺度分析、维数、组合学等基本概念足够深入，可以传达一些研究的味道，但可以用基本的方式成功地解释。研究成果将通过参加美国和国际会议进行传播，这将促进与现有和年轻研究人员的合作。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).With regards to the intellectual content of the project, the aim of this proposal is the study of interactions between quasiconformal (QC) mappings, geometric analysis (in particular uniform rectifiability), Fourier analysis, and geometric combinatorics. More specifically, the PI will pursue (1) Sharp distortion examples for K-QC maps. (2) Sharp sufficient conditions for removability of sets under bounded quasiregular (QR) maps (related to sharp QC distortion theorems.) (3) Buffon needle probability (Favard length) for Cantor sets (4) Questions involving uniformly rectifiable sets and harmonic measure. (5) Problems on distance sets relating Fourier analysis and geometric combinatorics. The unifying method to be employed is an underlying geometric-combinatorial vision which often manifests itself through multiscale analysis (i.e. the analysis of a problem on different scales.) This method will be applied in the contexts of K-QC mappings (mappings sending an infinitesimal circle/ball to an infinitesimal ellipse/ellipsoid with eccentricity controlled by K), geometric measure theory (GMT, which analyzes sets and measures on them -these are generalizations of length, area and volume-), harmonic analysis (decomposing a signal into elementary pieces of wavelike character), and potential theory (study of Coulombic potential and related topics.) With regards to contextualizing the proposed research within a broader mathematical and scientific framework, note that the mathematical objects involved have found abundant applications in other disciplines, so the problems proposed will advance knowledge in those areas and hence impact other areas of mathematics, science, or engineering. More specifically, fractals (geometric measure theory) appear naturally in electrodeposition and Diffusion Limited Aggregation. The internal structure of lungs has a high fractal dimension (to capture more oxygen.) Fourier analysis is often applied in signal and image processing. Quasiconformal maps are solutions to problems in non-linear elasticity, and have found applications in string theory. Uniform rectifiability appears in minimizers of the Mumford-Shah functional (originally used for image segmentation.) Geometric combinatorics is used for fair division and voting problems in the social sciences, and for phylogenetic trees models in biology. Distance sets are used in industry to study the dimensionality of data sets. In terms of human resource development, the PI will continue preparing students for the Putnam Competition, participating in the Math Club, and mentoring graduate students informally in the context of graduate courses. Fractals and geometric combinatorics are excellent areas for promoting teaching and training of undergraduates and postdocs. The basic notions of multiscale analysis, dimension, combinatorics, etc. are deep enough to convey some flavor of research yet can be successfully explained in an elementary way. Research results will be disseminated via participation in US and international meetings, which will facilitate collaborations with both established and young researchers.