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
- 来源:
- 关键词:
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的PI,克里斯托弗·毕晓普,将研究的几何性质的共形映射在平面和拟共形映射在空间中,侧重于这种映射的扩展性质和阐述各种应用几何函数理论,动力学和拓扑。 PI已经表明,结果丹尼斯沙利文's关于几何的凸体在双曲三空间意味着一个factorizationtheorem共形映射在平面上,这反过来又意味着统一的界限上的数额ofconstraction共形映射在平面上可以有。 寻找最佳常数的因式分解定理的后果众所周知的问题,如维数失真,积分手段和布伦南猜想。 PI将继续研究Kleinian群的极限集,Kleinian群是分形集的一个自然而重要的类别,这里的问题主要是估计这些集合的分维,并研究当群变形时维数的行为。PI还将研究调和测度的度量性质,特别是量化调和测度不能集中在一个小集合上的思想的结果。 问题包括低密度猜想、调和测度的稳定性和扩散限制聚集的增长率。 共形映射是一类在数学的许多领域都很重要的函数,它与许多物理问题(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多年。 这类地图的基本性质之一是膨胀;它们往往会使点平均距离更远。 使这一精确激发了数学分析中的许多研究。PI发现了一种新的方法来量化这种扩展,通过用(更一般的)拟共形映射来近似共形映射,并显示这些近似可能具有非常强的扩展性质。 这使我们对一些已知的结果有了更清楚的理解,并在新问题上取得了进展,特别是,这意味着关于Kleinian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、分形和混沌等更广泛的领域做出了贡献)。 PI的方法也将保形映射的行为与三维双曲空间的几何联系起来;这种联系似乎是新的,应该会导致许多有趣的问题,以及复分析和三维拓扑(已经以其他方式连接)领域之间的更多互动。 他还将研究这种连接的计算方面,这可能会导致计算共形映射和格林函数的新方法(对于各种应用很重要)。 PI还将继续他的调查其他问题,包括几何随机路径,如布朗运动,稳定性扰动下的某些动力系统和fundamentalgeometric性质的共形和quasiconformal映射。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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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
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
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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