Quasiconformal Constructions in Analysis and Dynamics

分析和动力学中的拟共形结构

• 批准号: 1906259
• 负责人: Christopher Bishop
• 金额: $ 26.91万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906259&HistoricalAwards=false
• 关键词: Quasiconformal; Constructions; Analysis; Dynamics

The project deals with various problems in 2 and 3 dimensional geometry by applying ideas from conformal and hyperbolic geometry to problems of computer science and statistical physics, and, conversely, utilizing ideas from discrete and computational geometry to study longstanding problems in complex analysis and dynamics. These methods, in addition to their intrinsic interest, provide new connections between discrete and continuous mathematics, as well as between theoretical and applied questions.This expands the range of tools available in all these areas, and facilitates more substantial interactions between disparate groups of mathematicians, physicists and computer scientists. The concrete, geometric, and practical aspects of these problems also make them appealing to students and can form the basis of student research projects. For example, what is the optimal way to decompose a 2 or 3 dimensional region as a union of triangles or tetrahedra? Doing this quickly (e.g., in polynomial time) with good control of the shapes of the mesh elements (e.g., no angles too large or too small) is a fundamental first step in most numerical modeling methods, and a good initial mesh leads to faster, more accurate solutions. Nevertheless, many meshing algorithms used in practice (essentially all used in three dimensions) are based on heuristics rather than rigorously justified methods. The PI's earlier work on meshing in 2 dimensions is based on exploiting ideas from conformal geometry and dynamics, and uses the PI's earlier work on optimal algorithms for computing conformal mappings. The PI plans to continue his work in 2 dimensions to find the optimal meshing algorithms and to seek analogous approaches in 3 dimensions. Conformal maps send one planar region to another in a way that preserves angles. Such maps have been studied for almost two hundred years and are fundamental to many areas including complex analysis, dynamics, fluid flow and probability. The connection with probability arises most strongly through Brownian motion, the mathematical model of continuous random motion. Conformal maps send Brownian paths to Brownian paths, and this `conformal invariance' allows probabilistic arguments to be applied in complex analysis and allows complex analysis to be used to study Brownian motion. The project considers another well known, but difficult to understand, random process related to conformal maps: diffusion limited aggregation (DLA for short). In DLA particles wander at random in space until they make contact with a fixed cluster of particles and stick to it. As time progresses the cluster grows and takes a characteristic fractal shape, at least in numerical simulations; very little is known rigorously about the geometry of the resulting random clusters. Based on numerical simulations, the project considers new conjectures on DLA that may be more tractable. The above problems seek to use classical analysis to understand problems that are discrete in nature. Another aspect of the project is to use discrete ideas to attack problems in continuous analysis. The PI previously introduced a new method, called quasiconformal folding, for constructing holomorphic maps that that emphasizes the discrete and combinatorial structure of holomorphic function. Using this method, the PI and other investigators have constructed examples that answered a number of open problems, including several in the field of holomorphic dynamics. This is the study of stable versus chaotic behavior when a function is iterated, e.g., the Julia set of a map is a fractal set where the sequence of iterates acts chaotically. Determining the possible geometries of this set is a fundamental problem, and the PI will work on specific conjectures the case of general holomorphic functions defined on the whole complex plane.The project consists of problems grouped into five main categories, but all dealing with the interaction of discrete combinatorial structures (usually trees or triangulations) with continuous problems related to complex analysis and conformal geometry (Riemann surfaces, holomorphic dynamics, hyperbolic 3-manifolds, computational geometry and statistical physics). The first part deals with conformal structures induced by trees and graphs. Grothendieck's theory of "dessins d'enfants" shows that a finite graph on a compact topological surface induces a conformal structure on that surface, e.g., makes it into a Riemann surface. Several problems in the project seek to understand this connection precisely, e.g., given a finite tree in the plane, what does the tree "look like" in the induced conformal structure? A related problem is to build a Riemann surface by identifying edges of equilateral triangles. It is known that not every compact surface occurs in this way (only countably many can occur), but the PI conjectures that every non-compact surface does occur in this way (joint work with Lasse Rempe-Gillen). The second group of problems deals with transcendental dynamics, i.e., the iteration theory of non-polynomial entire functions. Quasiconformal folding is a method of building entire functions, starting from an infinite planar tree, that gives very good control of the geometry of the function and locations of its critical values. The PI seeks to apply QC folding to various problems, such as the possible Hausdorff and packing dimension of transcendental Julia sets, and the behavior of these dimensions under perturbations of the function. The third part of the project is to expand on recent work of the PI and Claude LeBrun that constructed examples 4-manifolds where the almost-Kahler structures form a non-empty, proper open subset in the moduli space of ant-self-dual metric; such examples were not previously known to exist. The proof works by reducing to questions about harmonic measure on certain (very complicated) hyperbolic 3-manifolds. Can simpler 3-manifolds be used? Are the "exotic" examples constructed actually "common" in some sense? The fourth part deals with optimal meshing problems in 2 and 3 dimensions. The PI recently proved that any planar domain bounded by a PSLG (planar straight line graph) can be meshed by non-obtuse triangles (maximum angle 90 degrees) in polynomial time, but the optimal power remains open. The proof uses a discrete closing lemma for certain flows associated to any triangulation and perhaps ideas from surface dynamics might help close the gap. The proof uses several estimates from planar geometry, and it would be a very interesting and important to extend the results to triangulated surfaces, a stepping stone to the important problem of meshing 3-dimension regions by non-obtuse tetrahedra. A key idea in the planar case is to decompose the region into pieces where either Euclidean or hyperbolic geometry is used to generate a mesh. Can a similar strategy be used in three dimensions, perhaps using the wider variety of fundamental geometries that occur in three dimensions? Finally, the PI will consider diffusion limited aggregation (DLA), a widely studied random growth model. The basic problem is to determine the almost sure growth rate of the cluster; there is no non-trivial lower bound and no improvement of Kesten's 1987 upper bound. The PI plans break the growth rate problem into two sub-problems: estimating upper bounds for the number of vertices on the convex hull boundary of DLA, and turning these into non-trivial lower bounds for the growth rate.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.

该项目通过将保形几何和双曲几何的思想应用于计算机科学和统计物理的问题来处理二维和三维几何中的各种问题，反过来，利用离散几何和计算几何的思想来研究复杂分析和动力学中的长期问题。这些方法，除了它们固有的兴趣之外，还提供了离散数学和连续数学之间，以及理论和应用问题之间的新联系。这扩大了所有这些领域可用工具的范围，并促进了数学家、物理学家和计算机科学家等不同群体之间更实质性的互动。这些问题的具体、几何和实际方面也使它们对学生有吸引力，并可以形成学生研究项目的基础。例如，将二维或三维区域分解为三角形或四面体的并集的最佳方法是什么？快速完成此操作（例如，在多项式时间内）并良好地控制网格元素的形状（例如，没有太大或太小的角度）是大多数数值建模方法的基本第一步，良好的初始网格会导致更快，更准确的解决方案。然而，实践中使用的许多网格划分算法（基本上都是在三维中使用的）是基于启发式而不是严格证明的方法。PI在二维网格划分方面的早期工作是基于利用共形几何和动力学的思想，并使用PI在计算共形映射的最佳算法方面的早期工作。PI计划在二维中继续他的工作，以找到最佳的网格算法，并在三维中寻求类似的方法。保角映射以保持角度的方式将一个平面区域传送到另一个平面区域。这种地图已经被研究了近200年，是许多领域的基础，包括复杂分析、动力学、流体流动和概率。布朗运动是连续随机运动的数学模型，它与概率的联系最为密切。共形映射将布朗路径发送到布朗路径，这种“共形不变性”允许概率参数应用于复分析，并允许复分析用于研究布朗运动。该项目考虑了另一个众所周知的，但难以理解的，与保形映射相关的随机过程：扩散有限聚集（简称DLA）。在DLA中，粒子在空间中随机漫游，直到它们与一组固定的粒子接触并粘在一起。至少在数值模拟中，随着时间的推移，星团逐渐长大，呈现出典型的分形形状；对于由此产生的随机星团的几何结构，我们所知甚少。基于数值模拟，该项目考虑了可能更易于处理的关于DLA的新猜想。上述问题试图使用经典分析来理解本质上离散的问题。该项目的另一个方面是使用离散的思想来解决连续分析中的问题。PI先前介绍了一种新的方法，称为准共形折叠，用于构造全纯映射，强调全纯函数的离散和组合结构。利用这种方法，PI和其他研究人员已经构建了一些例子来回答一些开放的问题，包括全纯动力学领域的几个问题。这是对函数迭代时稳定与混沌行为的研究，例如，映射的Julia集是一个分形集，其中迭代序列是混沌的。确定这个集合的可能几何形状是一个基本问题，PI将处理在整个复平面上定义的一般全纯函数的特定猜想。该项目包括分为五个主要类别的问题，但都涉及离散组合结构（通常是树或三角）与复杂分析和共形几何（黎曼曲面，全纯动力学，双曲3流形，计算几何和统计物理）相关的连续问题的相互作用。第一部分讨论由树和图引起的共形结构。Grothendieck的“desdesins d'enfants”理论表明，紧致拓扑表面上的有限图在该表面上诱导出一个共形结构，例如使其成为黎曼曲面。项目中的几个问题试图精确地理解这种联系，例如，给定平面上的有限树，树在诱导保形结构中“看起来像”什么？一个相关的问题是通过识别等边三角形的边来建立黎曼曲面。众所周知，并不是每个紧致曲面都以这种方式发生（只有可数的几个可以发生），但PI猜想每个非紧致曲面都以这种方式发生（与Lasse Rempe-Gillen联合工作）。第二组问题涉及超越动力学，即非多项式整函数的迭代理论。拟共形折叠是一种从无限平面树开始构建整个函数的方法，它可以很好地控制函数的几何形状及其临界值的位置。PI试图将QC折叠应用于各种问题，例如超越Julia集的可能Hausdorff维数和包装维数，以及这些维数在函数扰动下的行为。该项目的第三部分是扩展PI和Claude LeBrun最近的工作，他们构建了4流形的例子，其中几乎kahler结构在反自对偶度量的模空间中形成了一个非空的，适当的开放子集；这样的例子以前并不知道存在。这个证明是通过简化到某些（非常复杂的）双曲3流形上的调和测度问题来进行的。可以使用更简单的3-流形吗？在某种意义上，这些“奇异的”例子实际上是“常见的”吗？第四部分是二维和三维的最优网格划分问题。PI最近证明了任何由PSLG（平面直线图）包围的平面域都可以在多项式时间内由非钝角三角形（最大角度为90度）网格化，但最优功率仍然是开放的。该证明使用了一个离散的闭合引理，用于与任何三角测量相关的某些流，也许来自表面动力学的想法可能有助于缩小差距。该证明使用了平面几何中的几个估计，将结果推广到三角曲面将是一个非常有趣和重要的问题，这是解决非钝角四面体网格化三维区域的重要问题的踏脚石。平面情况下的一个关键思想是将区域分解成块，其中欧几里得几何或双曲几何用于生成网格。类似的策略是否可以在三维空间中使用，也许可以使用三维空间中出现的更广泛的基本几何？最后，PI将考虑扩散有限聚集（DLA），一个广泛研究的随机增长模型。基本问题是确定集群的几乎确定的增长率；没有非平凡下界，也没有对1987年Kesten上界的改进。PI计划将增长率问题分解为两个子问题：估计DLA凸壳边界上顶点数的上界，并将其转化为增长率的非平凡下界。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。

项目成果

Optimal angle bounds for Steiner triangulations of polygons

多边形斯坦纳三角剖分的最佳角度范围

• 发表时间: 2022
• 期刊: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
• 作者: Bishop, ChristopherJ.

Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $\mathbb R^d$

Falconer 的 $(K, d)$ 距离集猜想对于 $mathbb R^d$ 中的严格凸集 $K$ 可能会失败

• DOI: 10.4171/rmi/1254
• 发表时间: 2021
• 期刊: Revista Matemática Iberoamericana
• 作者: Bishop, Christopher;Drillick, Hindy;Ntalampekos, Dimitrios
• 通讯作者: Ntalampekos, Dimitrios

Conformal images of Carleson curves

卡尔森曲线的共形图像

• DOI: 10.1090/bproc/69
• 发表时间: 2022
• 期刊: Series B
• 作者: Bishop, Christopher

Quasiconformal maps with thin dilatations

具有薄扩张的拟共形贴图

• DOI: 10.5565/publmat6622207
• 发表时间: 2022
• 期刊: Publicacions Matemàtiques
• 作者: Bishop, Christopher J.

Speiser class Julia sets with dimension near one

Speiser 级 Julia 集的维数接近 1

• DOI: 10.1007/s11854-020-0128-1
• 发表时间: 2020
• 期刊: Journal d'Analyse Mathématique
• 作者: Albrecht, Simon;Bishop, Christopher J.
• 通讯作者: Bishop, Christopher J.