Quasiconformal analysis, optimal triangulations and fractal geometry

该项目将使用经典的真实的和复杂的分析思想来解决计算机科学，物理学，动力学和概率中出现的问题。该项目的一个重点是优化网格和三角剖分，它有许多应用，从计算机图形到流体流动建模。在许多问题中，有必要将空间区域近似为三角形的有限并集。三角形的形状极大地影响各种数值方法的精度和速度，避免角度太小或太大是很重要的。在数值建模中，寻找良好的三角剖分通常是最耗时（且自动化程度最低）的步骤。对该项目的资助将导致算法的开发，该算法计算给定域的最佳角度范围，并找到实现这些范围的实用三角测量。 资助工作的其他部分将开发通过特殊嵌入到三维空间中来识别和分类二维形状的方法，并将改进用于模拟肿瘤和流体通过多孔材料传播的某些随机生长模型的估计。该项目为研究生提供了研究培训机会。这项工作将把PI为多边形平面域三角剖分引入的共形和拟共形方法扩展到涉及PSLG（平面直线图），非平面表面和三维区域的更困难的情况。 PI以前的工作涉及与三角剖分相关的动态流和这些流的封闭引理的离散类似物。这些结果将被推广和推广。 这项工作还将扩大PI的结果的光滑性和几何形状的Weil-Petersson曲线，帮助计算与这些曲线，这是密切相关的弦理论，模式识别，几何测度理论和随机曲线族。在这个项目中，平面曲线与双曲3-空间中的极小曲面相关联，使得曲线的几何性质反映在曲面的性质中。PI还将调查扩散限制聚合的增长估计，这是一个随机增长过程，来自统计物理学，受到许多自然过程的激励，但很少有严格的结果是已知的。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

The project will use ideas from classical real and complex analysis to solve problems arising in computer science, physics, dynamics and probability. One focus of the project is optimal meshing and triangulation, which has many applications, from computer graphics to modeling fluid flows. In many problems it is necessary to approximate a region of space as a finite union of triangles. The shapes of the triangles greatly influences the accuracy and speed of various numerical methods, and it is important to avoid angles that are too small or too large. Finding good triangulations is often the most time consuming (and least automated) step in numerical modeling. Funding for this project will lead to the development of algorithms that compute the best possible angle bounds for a given domain and find practical triangulations achieving these bounds. Other parts of the funded work will develop methods to recognize and classify 2-dimension shapes via special embeddings into 3-dimensional space, and will give improved estimates for certain random growth models that have been used to model tumors and the propagation of fluids through porous materials. The project provides research training opportunities for graduate students.This work will extend the conformal and quasiconformal methods introduced by the PI for triangulation of polygonal planar domains to more difficult situations involving PSLGs (planar straight line graphs), non-planar surfaces, and 3-dimensional regions. The PI's previous work involves dynamical flows associated to triangulations and discrete analogs of closing lemmas for these flows. These results will be extended and generalized. The work will also expand on the PI's results on the smoothness and geometry of Weil-Petersson curves, aiding computation with these curves, which are closely related to string theory, pattern recognition, geometric measure theory and random curve families. In this project, planar curves are associated to minimal surfaces in hyperbolic 3-space in such a way that geometric properties of the curve are reflected in the properties of the surface. The PI will also investigate growth estimates for diffusion limited aggregation, a random growth process from statistical physics that is motivated by a number of natural processes, but about which few rigorous results are known.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.