Geometry of Conformal and Quasiconformal Mappings

共形和拟共形映射的几何

基本信息

  • 批准号:
    0103626
  • 负责人:
  • 金额:
    $ 17.64万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2001
  • 资助国家:
    美国
  • 起止时间:
    2001-06-01 至 2005-05-31
  • 项目状态:
    已结题

项目摘要

Abstract for DMS - 0103626The PI, Christopher Bishop, will study the geometric propertiesof conformal mappings in the plane and quasiconformal mappings in space,focusing on the expansion properties of such maps and investigatingvarious applications to geometric function theory, dynamics andtopology. The PI has shown that a result of Dennis Sullivan'sconcerning the geometry of convexbodies in hyperbolic three space implies a factorizationtheorem for conformal mappings in the plane and this, in turn,implies uniform bounds on the amount ofcontraction a conformal map in the plane can have. Finding the best constantsin the factorization theorem has consequences for wellknown problems such as dimension distortion, integral means and Brennan'sconjecture. The PI will continue his work onlimit sets of Kleinian groups, a natural and important class of fractal sets.The questions here are mainly to estimate the fractal dimension of these setsand study the behavior of the dimension as the group is deformed. The PI will also work on the metric propertiesof harmonic measures, particularly results which quantify the idea that harmonicmeasure cannot be concentrated on a small set. Problems include thelower density conjecture, stability of harmonic measure and the growth rateof diffusion limited aggregation. A few other questionsinvolving quasiconformal and biLipschitz maps are also considered.Conformal mappings are a class of functions which are importantin many area of mathematics and which are closely relatedto mnay physical problems (fluid flow, heat conduction, electric fields, random growth models, ...) and have been intensively studied for many years. One of the fundamental properties of such maps is expansion; they tend to push points farther apart on average. Making this precise has motivated much research in mathematical analysis.The PI has discovered a new way of quantifyingthis expansion by approximating conformal maps by (the moregeneral class of) quasiconformal maps and showing theseapproximations may be taken with a very strong expansion property. This has given a clearer understanding of someknown results and has led to progress on new problems.In particular, it implies new results about Kleinian groups(these are important examples of conformal dynamical systems, and hence a contribution to the more general area of dynamical systems, fractals and chaos). The PI's approach also ties the behavior of conformal maps to the geometry three dimensional hyperbolic space; this connection seems to be new and should lead to many interesting problems and more interaction between the areas of complex analysis and three dimensional topology (already connected in other ways). He will also investigate the computationalaspects of this connection which may lead to new methods of computing conformal maps and Greens functions (importantfor a variety of applications). The PI will also continue his investigation of other problems including the geometry of randompaths such as Brownian motion, the stability underperturbation of certain dynamical systems and fundamentalgeometric properties of conformal and quasiconformal mappings.
摘要 DMS - 0103626 PI Christopher Bishop 将研究平面中的共形映射和空间中的拟共形映射的几何特性,重点关注此类映射的扩展特性,并研究在几何函数理论、动力学和拓扑中的各种应用。 PI 表明,丹尼斯·沙利文 (Dennis Sullivan) 关于双曲三空间中凸体几何形状的结果暗示了平面中共形映射的因式分解定理,而这又暗示了平面中共形映射可以具有的收缩量的统一界限。 寻找因式分解定理中的最佳常数对于维数畸变、积分均值和布伦南猜想等众所周知的问题具有重要意义。 PI 将继续他对克莱因群极限集的研究,克莱因群是一类自然而重要的分形集。这里的问题主要是估计这些集合的分形维数,并研究该维数在群变形时的行为。 PI 还将研究调和测量的度量属性,特别是量化调和测量不能集中在一个小集合上的想法的结果。 问题包括低密度猜想、调和测度的稳定性以及扩散限制聚集的增长率。 还考虑了涉及拟共形和biLipschitz映射的其他一些问题。共形映射是一类在许多数学领域中都很重要的函数,并且与许多物理问题(流体流动、热传导、电场、随机增长模型等)密切相关,并且已经被深入研究了很多年。 此类地图的基本属性之一是扩展性。他们倾向于将平均点拉得更远。 使其精确化激发了数学分析方面的大量研究。PI 发现了一种量化这种展开的新方法,通过(更一般的一类)拟共形映射来近似共形映射,并显示这些近似可以具有非常强的展开特性。 这使得人们对一些已知的结果有了更清晰的理解,并在新问题上取得了进展。特别是,它意味着关于克莱因群的新结果(这些是共形动力系统的重要例子,因此对动力系统、分形和混沌等更普遍的领域做出了贡献)。 PI 的方法还将共形映射的行为与几何三维双曲空间联系起来;这种联系似乎是新的,应该会导致许多有趣的问题以及复杂分析和三维拓扑领域之间的更多交互(已经以其他方式连接)。 他还将研究这种连接的计算方面,这可能会带来计算共形图和格林函数的新方法(对于各种应用都很重要)。 PI 还将继续研究其他问题,包括随机路径的几何形状(例如布朗运动)、某些动力系统的稳定性欠扰动以及共形和拟共形映射的基本几何特性。

项目成果

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Christopher Bishop其他文献

The novel analog 1,24(S)-dihydroxyvitamin D2 is as equipotent as 1,25-dihydroxyvitamin D3 in growth regulation of cancer cell lines.
新型类似物 1,24(S)-二羟基维生素 D2 在癌细胞系生长调节方面与 1,25-二羟基维生素 D3 等效。
  • DOI:
  • 发表时间:
    1998
  • 期刊:
  • 影响因子:
    2
  • 作者:
    Y. Levy;Knutson Jc;Christopher Bishop;S. Shany
  • 通讯作者:
    S. Shany
Exploring Gender Roles and Gender Equality within the Evangelical Church
探索福音派教会内的性别角色和性别平等
  • DOI:
    10.36837/chapman.000037
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    2.1
  • 作者:
    Christopher Bishop
  • 通讯作者:
    Christopher Bishop
DeepSpeed4Science Initiative: Enabling Large-Scale Scientific Discovery through Sophisticated AI System Technologies
DeepSpeed4Science 计划:通过复杂的人工智能系统技术实现大规模科学发现
  • DOI:
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    S. Song;Bonnie Kruft;Minjia Zhang;Conglong Li;Shiyang Chen;Chengming Zhang;Masahiro Tanaka;Xiaoxia Wu;Jeff Rasley;A. A. Awan;Connor Holmes;Martin Cai;Adam Ghanem;Zhongzhu Zhou;Yuxiong He;Christopher Bishop;Max Welling;Tie;Christian Bodnar;Johannes Brandsetter;W. Bruinsma;Chan Cao;Yuan Chen;Peggy Dai;P. Garvan;Liang He;E. Heider;Pipi Hu;Peiran Jin;Fusong Ju;Yatao Li;Chang Liu;Renqian Luo;Qilong Meng;Frank Noé;Tao Qin;Janwei Zhu;Bin Shao;Yu Shi;Wen;Gregor Simm;Megan Stanley;Lixin Sun;Yue Wang;Tong Wang;Zun Wang;Lijun Wu;Yingce Xia;Leo Xia;Shufang Xie;Shuxin Zheng;Jianwei Zhu;Pete Luferenko;Divya Kumar;Jonathan Weyn;Ruixiong Zhang;Sylwester Klocek;V. Vragov;Mohammed Alquraishi;Gustaf Ahdritz;C. Floristean;Cristina Negri;R. Kotamarthi;V. Vishwanath;Arvind Ramanathan;Sam Foreman;Kyle Hippe;T. Arcomano;R. Maulik;Max Zvyagin;Alexander Brace;Bin Zhang;Cindy Orozco Bohorquez;Austin R. Clyde;B. Kale;Danilo Perez;Heng Ma;Carla M. Mann;Michael Irvin;J. G. Pauloski;Logan Ward;Valerie Hayot;M. Emani;Zhen Xie;Diangen Lin;Maulik Shukla;Thomas Gibbs;Ian Foster;James J. Davis;M. Papka;Thomas Brettin;Prasanna Balaprakash;Gina Tourassi;John P. Gounley;Heidi Hanson;T. Potok;Massimiliano Lupo Pasini;Kate Evans;Dan Lu;D. Lunga;Junqi Yin;Sajal Dash;Feiyi Wang;M. Shankar;Isaac Lyngaas;Xiao Wang;Guojing Cong;Peifeng Zhang;Ming Fan;Siyan Liu;A. Hoisie;Shinjae Yoo;Yihui Ren;William Tang;K. Felker;Alexey Svyatkovskiy;Hang Liu;Ashwin Aji;Angela Dalton;Michael Schulte;Karl Schulz;Yuntian Deng;Weili Nie;Josh Romero;Christian Dallago;Arash Vahdat;Chaowei Xiao;Anima Anandkumar;R. Stevens
  • 通讯作者:
    R. Stevens
Effects of genetic knockdown of the serotonin transporter on established L-DOPA-induced dyskinesia and gene expression in hemiparkinsonian rats
5-羟色胺转运体基因敲除对已建立的左旋多巴诱导的帕金森病大鼠运动障碍及基因表达的影响
  • DOI:
    10.1016/j.neuropharm.2024.110227
  • 发表时间:
    2025-03-15
  • 期刊:
  • 影响因子:
    4.600
  • 作者:
    Grace McManus;Ashley Galfano;Carla Budrow;Natalie Lipari;Kuei Y. Tseng;Fredric P. Manfredsson;Christopher Bishop
  • 通讯作者:
    Christopher Bishop
The burden of the present in Gareth Brookes, The Dancing Plague
加雷斯·布鲁克斯《跳舞的瘟疫》中当下的负担
  • DOI:
    10.1080/1472586x.2022.2050101
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0.7
  • 作者:
    Christopher Bishop
  • 通讯作者:
    Christopher Bishop

Christopher Bishop的其他文献

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{{ truncateString('Christopher Bishop', 18)}}的其他基金

Quasiconformal analysis, optimal triangulations and fractal geometry
拟共形分析、最优三角剖分和分形几何
  • 批准号:
    2303987
  • 财政年份:
    2023
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Standard Grant
I-Corps: Repurposing Serotoninergic Compounds for Improved Treatment of Parkinson's Disease
I-Corps:重新利用血清素能化合物以改善帕金森病的治疗
  • 批准号:
    2148598
  • 财政年份:
    2021
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Standard Grant
Quasiconformal Constructions in Analysis and Dynamics
分析和动力学中的拟共形结构
  • 批准号:
    1906259
  • 财政年份:
    2019
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Continuing Grant
Geometric Problems in Conformal Analysis, Dynamics, and Probability
共形分析、动力学和概率中的几何问题
  • 批准号:
    1608577
  • 财政年份:
    2016
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Continuing Grant
Quasiconformal methods in analysis, geometry and dynamics
分析、几何和动力学中的拟共形方法
  • 批准号:
    1305233
  • 财政年份:
    2013
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Continuing Grant
Analysis of conformal and quasiconformal maps
共形和拟共形映射的分析
  • 批准号:
    1006309
  • 财政年份:
    2010
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Standard Grant
Computational and Conformal Geometry
计算和共形几何
  • 批准号:
    0705455
  • 财政年份:
    2007
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Continuing Grant
Geometry of Conformal and Quasiconformal Mappings
共形和拟共形映射的几何
  • 批准号:
    0405578
  • 财政年份:
    2004
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Standard Grant
Deformations of Complex Structures
复杂结构的变形
  • 批准号:
    9800924
  • 财政年份:
    1998
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Postdoctoral Research Fellowship
数学科学:博士后研究奖学金
  • 批准号:
    8705957
  • 财政年份:
    1987
  • 资助金额:
    $ 17.64万
  • 项目类别:
    Fellowship Award

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量子引力中的非微扰共形场论和实验室(精确 CFT)
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