Quasiconformal analysis, optimal triangulations and fractal geometry

该项目将使用经典实分析和复杂分析的思想来解决计算机科学、物理学、动力学和概率论中出现的问题。该项目的一个重点是优化网格和三角测量，它有许多应用，从计算机图形学到流体流动建模。在许多问题中，有必要将空间区域近似为三角形的有限并。三角形的形状对各种数值方法的精度和速度有很大的影响，避免角度过小或过大是很重要的。在数值建模中，寻找好的三角剖分通常是最耗时的（也是最不自动化的）步骤。该项目的资助将导致算法的发展，计算给定域的最佳角度边界，并找到实现这些边界的实用三角测量。该基金的其他部分工作将开发通过特殊嵌入到三维空间来识别和分类二维形状的方法，并将对某些用于模拟肿瘤和流体通过多孔材料传播的随机生长模型给出改进的估计。本项目为研究生提供研究训练机会。这项工作将把PI引入的用于多边形平面域三角剖分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测量相关的动态流和这些流的闭合引理的离散类似物。这些结果将得到推广和推广。这项工作还将扩展PI在Weil-Petersson曲线的平滑性和几何形状方面的结果，帮助这些曲线的计算，这些曲线与弦理论、模式识别、几何测量理论和随机曲线族密切相关。在这个项目中，平面曲线与双曲三维空间中的最小曲面相关联，曲线的几何属性反映在曲面的属性中。PI还将研究扩散有限聚集的增长估计，这是统计物理学中的一种随机增长过程，由许多自然过程驱动，但很少有严格的结果已知。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.