Geometry of Conformal and Quasiconformal Mappings

共形和拟共形映射的几何

基本信息

  • 批准号:
    0405578
  • 负责人:
  • 金额:
    $ 20万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2004
  • 资助国家:
    美国
  • 起止时间:
    2004-08-15 至 2007-07-31
  • 项目状态:
    已结题

项目摘要

AbstractAward: DMS-0405578Principal Investigator: Christopher BishopThe PI, Christopher Bishop, will study the geometric propertiesof conformal and quasiconformal mappings, focusing on theinteractions between conformal structures, hyperbolic geometry,low dimensional topology and numerical analysis. For example,Moore's theorem states that if we take a certain collection ofsets on the 2-sphere and topologically collapse them to pointsthen we obtain a new topological sphere. The PI will investigatewhen the quotient map can be conformal and will consider a numberof concrete problems from this general perspective, includingconformal welding, characterizations of John domains, Koebe'sconjecture, construction of Kleinian groups and other dynamicalobjects. The PI will continue his earlier work on the geometryof Kleinian limit sets and the behavior of the dimension as wedeform the limit set. The PI will also continue his work on theconnections between computational geometry, hyperbolic geometryand conformal mappings, and seek new algorithms which compute theRiemann mapping quickly and with rigorous error estimates. Inparticular he will investigate computing conformal maps using themedial axis (an object from computational geometry) which isclosely linked to 3-dimensional hyperbolic geometry via convexhulls.Conformal mappings are important both for their central role innumerous mathematical problems (complex analysis, dynamicalsystems,...) and in various applications (fluid flow, brainmapping, statistical physics, numerical analysis of differentialequations,...), so we must have a good theoretical understandingof these maps and good methods for computing them inpractice. The proposal deepens our theoretical understanding ofconformal maps by investigating new connections with other partsof mathematics and computer science (point set topology,3-dimensional hyperbolic geometry, Voronoi diagrams) and seeks touse these connections to invent new algorithms for computingconformal maps. For example, the medial axis is a widely studiedobject in computer science (it is a description of the shape onan object which has numerous applications in pattern recognition,robotic motion, biology,...). The PI discovered it can also beused to give a rough but fast approximation to conformal maps.The PI will seek to improve this method, giving better algorithmsfor conformal maps and also developing a better understanding ofthe medial axis which will impact its other applications (forexample, the medial axis can change drastically when the objectbeing described changes only a little; this is a seriouscomputational problem which can be addressed by thinking of themedial axis as an object in hyperbolic geometry instead of theusual Euclidean geometry). The connection between the medial axisand conformal maps may also lead to new ideas for studyingapplications in three dimensions (where conformal mappings do notexist, but maps based on the medial axis still do). Threedimensional problems are the most important for applications, butour understanding lags far behind the two dimensional case, sonew two dimensional ideas which generalize to higher dimensionsare important.
摘要奖项:DMS-0405578 首席研究员:Christopher Bishop PI Christopher Bishop 将研究共形和拟共形映射的几何性质,重点关注共形结构、双曲几何、低维拓扑和数值分析之间的相互作用。 例如,摩尔定理指出,如果我们在 2-球体上采用特定的集合集合并将它们拓扑折叠为点,那么我们将获得一个新的拓扑球体。 PI 将研究商图何时可以共形,并从这个一般角度考虑许多具体问题,包括共形焊接、John 域的表征、Koebe 猜想、Kleinian 群的构造和其他动态对象。 PI 将继续他早期关于克莱因极限集的几何学和变形极限集时尺寸行为的工作。 PI 还将继续研究计算几何、双曲几何和共形映射之间的联系,并寻求快速计算黎曼映射并具有严格误差估计的新算法。 特别是,他将研究使用中轴(计算几何中的一个对象)计算共形映射,该映射通过凸包与 3 维双曲几何密切相关。共形映射因其在众多数学问题(复杂分析、动力系统等)中的核心作用以及在各种应用(流体流动、脑图绘制、统计物理、数值分析)中的核心作用而非常重要。 微分方程,...),因此我们必须对这些映射有良好的理论理解,并在实践中计算它们的良好方法。该提案通过研究与数学和计算机科学其他部分(点集拓扑、3 维双曲几何、Voronoi 图)的新联系,加深了我们对共形图的理论理解,并寻求利用这些联系来发明计算共形图的新算法。 例如,中轴是计算机科学中广泛研究的对象(它是对对象形状的描述,在模式识别、机器人运动、生物学等领域有大量应用)。 PI 发现它还可以用于对共形图进行粗略但快速的近似。PI 将寻求改进这种方法,为共形图提供更好的算法,并更好地理解中轴,这将影响其其他应用(例如,当所描述的对象仅发生一点变化时,中轴可能会发生巨大变化;这是一个严重的计算问题,可以通过将中轴视为一个对象来解决) 在双曲几何中而不是通常的欧几里得几何中)。中轴和共形映射之间的联系也可能会带来研究三维应用的新想法(共形映射不存在,但基于中轴的映射仍然存在)。三维问题对于应用来说是最重要的,但我们的理解远远落后于二维情况,因此推广到更高维度的新二维思想很重要。

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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    $ 20万
  • 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:
    0103626
  • 财政年份:
    2001
  • 资助金额:
    $ 20万
  • 项目类别:
    Continuing 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:
    $ 20万
  • 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:
    $ 20万
  • 项目类别:
    Fellowship Award

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