Uniformization of non-uniform geometries

近几十年来，人们对分形和表现出非均匀几何形状的随机物体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，随机性是指这些模式中的不确定性。这里的非均匀几何意味着给定物体的几何特征在所有位置和尺度上都不遵循相同的定量标准。在数学中，分形研究的实质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，在天线设计，地形分析和目标识别。这个项目围绕着关于表面均匀化和同时保形焊接的问题，特别是在随机性存在的情况下。首席研究员将适应和扩展保形均匀化和焊接技术到新的非均匀和分形动态设置。研究者还打算指导不同层次的学生，从高中到博士，并在积极发展的分形几何和动力学领域促进研究。在这个项目中考虑的几何上不均匀的对象是度量曲线和空间，它们不准对称地等同于圆、等边三角形等。这种空间的例子在二次多项式动力学、保形焊接问题和随机均匀化中都很丰富。该项目的一个具体目标是引入一类新的分布在球面上的随机曲面，并考虑这类曲面的类型问题。在这种情况下，当这些曲面（几乎肯定）是抛物线，将研究相关函数的值分布性质，特别是这些函数的增长顺序。另一个目标是开发同步的保形焊接工具和技术，从而扩展最近在将反射组与严格固定的反有理图合并方面的结果。所采用的方法本质上是复解析的。该项目的成功将对随机表面均匀化和保形焊接方面不断增长的文献做出新的贡献，将提出随机值分布方面的新问题，并将开发超越当前方法所使用的新工具。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。

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.