CAREER: Weighted Inequalities and their Applications to Quasiconformal Maps

本提案的目的是研究拟共形（QC）映射、几何分析（特别是均匀可纠偏性）、傅立叶分析和几何组合之间的相互作用。解决五个问题。要采用的统一方法是一种潜在的几何组合视觉，这种视觉通常通过多尺度分析（即在不同尺度上分析问题）表现出来。这种方法将应用于所谓的“K-QC映射”（将无穷小的圆/球映射到无穷小的椭圆/椭球），几何测量理论（GMT，分析它们上的集合和度量-这些是长度，面积和体积的概括-），谐波分析（将信号分解为波状特征的基本片段）和势理论（研究库仑势和相关主题）。所涉及的数学对象在其他学科中有广泛的应用，因此提出的问题将促进这些领域的知识。分形（GMT）在电沉积和扩散有限聚集中自然出现。肺的内部结构具有很高的分形维数（以捕获更多的氧气）。傅里叶分析经常应用于信号和图像处理。QC映射是非线性弹性问题的解决方案，并在弦理论中得到了应用。均匀可整流性出现在Mumford-Shah函数（最初用于图像分割）的最小化中。几何组合用于社会科学中的公平划分和投票问题，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。PI将继续为学生准备普特南竞赛，参加数学俱乐部，并在研究生课程的背景下对研究生进行非正式指导。分形学和几何组合学是促进本科生和博士后教学和培养的极好领域。多尺度分析、维数、组合等基本概念足够深刻，可以传达一些研究的味道，但可以用一种基本的方式成功地解释。

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. Five problems will be addressed. 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 so called "K-QC mappings" (mappings sending an infinitesimal circle/ball to an infinitesimal ellipse/ellipsoid), 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.) The mathematical objects involved have found abundant applications in other disciplines, so the problems proposed will advance knowledge in those areas. Fractals (GMT) 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. QC 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. 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.