CAREER: Weighted Inequalities and their Applications to Quasiconformal Maps

职业：加权不等式及其在拟共形映射中的应用

• 批准号: 1056965
• 负责人: Ignacio Uriarte-Tuero
• 金额: $ 40.33万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1056965&HistoricalAwards=false
• 关键词: CAREER; Weighted; Inequalities; their; Applications

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.

该提案的目的是研究拟共形 (QC) 映射、几何分析（特别是均匀可校正性）、傅立叶分析和几何组合之间的相互作用。将解决五个问题。要采用的统一方法是一种基本的几何组合视觉，它通常通过多尺度分析（即在不同尺度上分析问题）来体现。该方法将应用于所谓的“K-QC映射”（将无穷小圆/球发送到无穷小椭圆/椭圆体的映射）、几何测度理论（GMT，它分析 集及其测量（这些是长度、面积和体积的概括）、调和分析（将信号分解为波状特征的基本片段）和势能理论（库仑势和相关主题的研究）。所涉及的数学对象在其他学科中已经找到了丰富的应用，因此提出的问题将推进这些领域的知识。分形 (GMT) 在电沉积和扩散有限聚集中自然出现。肺部的内部结构具有较高的分形维数（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中。 QC 图是非线性弹性问题的解决方案，并在弦理论中得到了应用。一致可校正性出现在 Mumford-Shah 函数的最小化中（最初用于图像分割）。几何组合学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训的绝佳领域。多尺度分析、维度、组合学等的基本概念足够深入，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。