Uniformization of non-uniform geometries

近几十年来，人们对分形和随机物体的非均匀几何形状的理解越来越感兴趣。分形是具有一定程度的自相似性或粗糙度或两者兼而有之的空间，随机性指的是这些模式的不确定性。这里的非均匀几何意味着给定对象的几何特征在所有位置和尺度上不遵守相同的定量标准。在数学中，对分形研究的浓厚兴趣来自于它们在动力系统理论中的应用。在数学之外，分形有许多应用，例如天线设计，地形分析和目标识别。该项目围绕着表面均匀化和同时保形焊接的问题和难题，特别是在随机性的存在下。主要研究人员将适应和扩展共形均匀化和焊接技术，以新的非均匀和分形动态设置。调查人员还打算指导从高中到博士的各个层次的学生，促进分形几何和动力学领域的研究。在这个项目中考虑的几何非均匀对象是度量曲线和空间，不拟对称相当于圆，等边三角形，等这样的空间的例子是丰富的动态二次多项式，保形焊接问题，并在随机均匀化。该项目的一个具体目标是引入一类新的随机曲面分布在球体上，并考虑这类曲面的类型问题。在这种情况下，当这样的表面是（几乎肯定）抛物线，相关的功能，特别是这种功能的增长顺序的值分布特性，将进行调查。另一个目标是开发同时保形焊接工具和技术，反过来将扩展最近的结果合并反射组与严格固定的反理性地图。所采用的方法是复杂的分析性质。该项目的成功将为随机表面均匀化和保形焊接的不断增长的文献做出新的贡献，将提出随机值分布的新问题，并将开发超越当前方法的新工具。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Recent decades have witnessed increased interest in the understanding of fractal and random objects that exhibit non-uniform geometries. Fractals are spaces that possess either a degree of self-similarity, or roughness, or both, and randomness refers to uncertainty in these patterns. Non-uniform geometry here means that the geometric features of the given objects do not adhere to the same quantitative standards across all locations and scales. Within mathematics, substantial interest in the study of fractals comes from their applications in the theory of dynamical systems. Outside of mathematics, fractals have numerous applications, for instance, in antenna design, terrain analysis, and target recognition. This project revolves around questions and problems regarding surface uniformization and simultaneous conformal welding, in particular, in the presence of randomness. The principal investigator will adapt and extend conformal uniformization and welding techniques to new non-uniform and fractal dynamical settings. The investigator also intends to mentor students at various levels, from high school to Ph.D., and foster research in the actively evolving field of fractal geometry and dynamics. The geometrically non-uniform objects considered in this project are metric curves and spaces that are not quasisymmetrically equivalent to circles, equilateral triangulations, etc. Examples of such spaces are abundant in the dynamics of quadratic polynomials, in conformal welding problems, and in random uniformization. One specific goal of the project is to introduce a new class of random surfaces spread over the sphere and to consider the type problem for surfaces of this class. In the case when such surfaces are (almost surely) parabolic, the value distribution properties of associated functions, particularly the order of growth of such functions, will be investigated. Another goal is to develop simultaneous conformal welding tools and techniques that in turn will extend recent results on merging reflection groups with critically fixed anti-rational maps. The methods to be employed are complex analytic in nature. Success in this project will result in new contributions to the growing literature on random surface uniformization and conformal welding, will raise new questions in random value distribution, and will develop new tools which go beyond those employed in current methodology.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.