Uniformization and Rigidity in Metric Surfaces and in the Complex Plane
公制曲面和复平面中的均匀化和刚度
基本信息
- 批准号:2246485
- 负责人:
- 金额:$ 23.98万
- 依托单位:
- 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:2023-07-01 至 2026-06-30
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
In this project, the PI aims to develop techniques for the deeper understanding of fractals; that is, objects whose shape is not smooth and potentially have cusps and wrinkles, or objects with possibly self-similar repeating patterns. Such objects appear in nature as coastlines, mountainous landscapes, river networks, lightning bolts, snowflakes, growth models of plants and crystals, and soap films. The questions the PI plans to study have applications whenever storage of three-dimensional information (landscapes, faces, human brain surface) in a two-dimensional image is desired without loss of information. While in the case of smooth objects (the opposite of fractals) the corresponding mathematical theory is well understood, this is not the case for fractal objects, which require the development of new techniques. Another focus of this project is on rigidity problems, asking whether it is possible to deform a fractal object that is made out of a flexible material into another fractal object, with controlled distortion. Also, fractal sets appear sometimes as boundaries of otherwise smooth objects; another rigidity problem concerns whether these fractals are removable, in the sense that their presence can be ignored for transformation purposes. Rigidity problems on fractal sets have applications in mathematical problems that require "gluing" together two functions, or two dynamical systems, or two surfaces, and could result in the better understanding of dynamical systems in physics. This project will also incorporate the training and professional development of graduate students. The main focus of the project is on two interrelated types of problems on fractals: uniformization and rigidity problems. The uniformization problem asks for geometric conditions on a fractal metric space so that it can be transformed to a smooth space with a well-behaved transformation that preserves the geometry, such as quasiconformal or quasisymmetric maps. Major progress has been made recently towards the quasiconformal uniformization problem with the involvement of the PI. The current project expects to develop an analytic theory for two-dimensional surfaces of locally finite area under no other assumption; the classical approaches in the field of analysis on metric spaces require instead several additional and restrictive geometric assumptions. Specifically, the PI will study the quasiconformal classification of non-smooth surfaces, the embedding of fractal surfaces in Euclidean space, the uniformization of 2-dimensional spheres of infinite area, and potential theory on fractal surfaces. Regarding rigidity problems, the PI will work on the problem of conformal removability, which asks whether a given compact subset of Euclidean space is negligible from the domain of a conformal map. The PI in recent works has displayed several new examples of removable and non-removable planar sets and has found a striking connection between the problems of uniformization and removability. Moreover, the PI has identified a new general class of sets that he conjectures to provide a characterization of removable sets. The PI will study this conjecture, as well as several related removability and rigidity problems in complex dynamics, geometric group theory, and circle domains.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.
在这个项目中,PI 的目标是开发能够更深入地理解分形的技术;也就是说,形状不光滑并且可能有尖点和皱纹的物体,或者可能具有自相似重复图案的物体。这些物体在自然界中表现为海岸线、山地景观、河流网络、闪电、雪花、植物和晶体的生长模型以及肥皂膜。每当需要在二维图像中存储三维信息(风景、面孔、人脑表面)且不丢失信息时,PI 计划研究的问题都可以应用。虽然在光滑物体(与分形相反)的情况下,相应的数学理论是很好理解的,但分形物体的情况并非如此,这需要开发新技术。该项目的另一个重点是刚性问题,询问是否有可能将由柔性材料制成的分形物体变形为另一个分形物体,并具有受控的变形。此外,分形集有时会出现为平滑对象的边界。另一个刚性问题涉及这些分形是否可去除,即出于变换目的可以忽略它们的存在。分形集上的刚性问题可应用于需要将两个函数、两个动力系统或两个表面“粘合”在一起的数学问题,并且可以更好地理解物理学中的动力系统。该项目还将纳入研究生的培训和专业发展。该项目的主要焦点是分形上两种相互关联的问题:均匀化和刚性问题。均匀化问题要求分形度量空间上的几何条件,以便可以通过保留几何形状的良好变换将其变换为平滑空间,例如拟共形或拟对称映射。最近,在 PI 的参与下,准共形均匀化问题取得了重大进展。当前项目期望在没有其他假设的情况下开发局部有限区域二维表面的解析理论;相反,度量空间分析领域的经典方法需要一些额外的和限制性的几何假设。具体来说,PI将研究非光滑表面的拟共形分类、分形表面在欧几里得空间中的嵌入、无限面积的二维球体的均匀化以及分形表面的势理论。关于刚性问题,PI 将研究共形可去除性问题,该问题询问欧几里得空间的给定紧子集是否在共形映射的域中可以忽略不计。 PI 在最近的工作中展示了几个可移动和不可移动平面集的新例子,并发现了均匀化和可移动性问题之间的惊人联系。此外,PI 还确定了一个新的通用集合类,他推测该集合可提供可移除集合的表征。 PI 将研究这一猜想,以及复杂动力学、几何群论和圆域中的几个相关的可移除性和刚性问题。该奖项反映了 NSF 的法定使命,并通过使用基金会的智力价值和更广泛的评估进行评估,认为值得支持。影响审查标准。
项目成果
期刊论文数量(0)
专著数量(0)
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会议论文数量(0)
专利数量(0)
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Dimitrios Ntalampekos其他文献
Semi-hyperbolic rational maps and size of Fatou components
半双曲有理图和 Fatou 分量的大小
- DOI:
- 发表时间:
2017 - 期刊:
- 影响因子:0
- 作者:
Dimitrios Ntalampekos - 通讯作者:
Dimitrios Ntalampekos
On the inverse absolute continuity of quasiconformal mappings on hypersurfaces
超曲面上拟共形映射的逆绝对连续性
- DOI:
- 发表时间:
2018 - 期刊:
- 影响因子:1.7
- 作者:
Dimitrios Ntalampekos;Matthew Romney - 通讯作者:
Matthew Romney
Non-removability of the Sierpiński gasket
Sierpiński 垫圈的不可拆卸性
- DOI:
10.1007/s00222-018-00852-3 - 发表时间:
2019 - 期刊:
- 影响因子:3.1
- 作者:
Dimitrios Ntalampekos - 通讯作者:
Dimitrios Ntalampekos
Lipschitz-Volume Rigidity and Sobolev Coarea Inequality for Metric Surfaces
公制曲面的 Lipschitz 体积刚度和 Sobolev 面积不等式
- DOI:
10.1007/s12220-024-01577-x - 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
D. Meier;Dimitrios Ntalampekos - 通讯作者:
Dimitrios Ntalampekos
A removability theorem for Sobolev functions and detour sets
Sobolev 函数和绕行集的可移性定理
- DOI:
10.1007/s00209-019-02405-7 - 发表时间:
2017 - 期刊:
- 影响因子:0.8
- 作者:
Dimitrios Ntalampekos - 通讯作者:
Dimitrios Ntalampekos
Dimitrios Ntalampekos的其他文献
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{{ truncateString('Dimitrios Ntalampekos', 18)}}的其他基金
Uniformization of Metric Spaces and Quasiconformal Removability
度量空间的均匀化和拟共形可去除性
- 批准号:
2000096 - 财政年份:2020
- 资助金额:
$ 23.98万 - 项目类别:
Standard Grant
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