Uniformization of non-uniform geometries

非均匀几何形状的均匀化

• 批准号: 2247364
• 负责人: Sergiy Merenkov
• 金额: $ 22.37万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247364&HistoricalAwards=false
• 关键词: Uniformization; non; uniform; geometries

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.

最近的几十年目睹了对表现出不均匀几何形状的分形和随机物体的理解的兴趣。分形是具有一定程度的自相似性或粗糙度或两者的空间，并且随机性是指这些模式的不确定性。这里的不均匀几何形状意味着给定对象的几何特征并不遵守所有位置和尺度上相同的定量标准。在数学中，对分形的研究的重大兴趣来自它们在动态系统理论中的应用。在数学之外，分形具有许多应用，例如在天线设计，地形分析和目标识别中。该项目围绕着关于表面均匀化和同时焊接的问题和问题，尤其是在存在随机性的情况下。主要研究者将适应并将保形均匀化和焊接技术扩展到新的非均匀和分形动力学设置。研究人员还打算在高中到博士学位的各个级别的指导学生，并在分形几何学和动力学的积极发展领域进行研究。该项目中考虑的几何不均匀的物体是度量曲线和空间，这些空间在准对称上等同于圆，等边三角形等。此类空间的示例在二次多项式的动力学中很丰富，在互构焊接问题的动力学中，以及在随机均匀的情况下。该项目的一个具体目标是引入一类新的随机表面，分布在球体上，并考虑此类表面的类型问题。在此类表面（几乎肯定）抛物线的情况下，将研究相关功能的价值分布特性，尤其是此类功能的增长顺序。另一个目标是开发同时使用的保形焊接工具和技术，而这些工具和技术反过来将扩展有关将反射组与截然不同的反理性图合并的最新结果。所使用的方法本质上是复杂的分析。该项目的成功将为不断增长的关于随机表面均匀化和保形焊接的文献做出新的贡献，将在随机价值分配中提出新的问题，并将开发出超越当前方法中使用的工具。该奖项反映了NSF的法定任务，并通过评估该基金会的智力优点和广泛的影响来评估NSF的法定任务。