Topology and Dynamics of Geometric Structures
几何结构的拓扑和动力学
基本信息
- 批准号:1709791
- 负责人:
- 金额:$ 43.29万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2017
- 资助国家:美国
- 起止时间:2017-07-01 至 2023-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
This project concerns the classification of geometric structures on manifolds. A "geometric structure" means a set of coordinate systems where the coordinates live in a space with a "classical geometry:" examples include Euclidean geometry, non-Euclidean geometry, projective geometry, affine geometry or conformal geometry. Since the mid 19th century, considerations of symmetry emphasize the viewpoint that a classical geometry is just the study of properties which are unchanged under the transitive action of a Lie group. Closely related is the importance of symmetry in modern physics, and many of objects in this investigation have physical motivations. Classification of geometric structures on a fixed topology means putting the more rigid geometric measurements on a loosely organized collection of points, which is the topology. A simple example is the fact that the sphere cannot support Euclidean geometry. This is just the mathematical abstraction of the fact that there is no metrically accurate world atlas. At least one page of the world atlas will distort distance. On the other hand the torus (the surface of a doughnut, bagel or inner tube) does admit Euclidean structures. In fact the different ways of putting Euclidean geometry on a torus has a rich geometry of its own. Technology plays a key tool both in this investigation and its dissemination to the public. Many mathematicians are involved in this project, including colleagues, postdocs, graduate students, undergraduate students and even high school students. The objects of study are often two- and three-dimensional and can be visualized with the use of computers. All the software developed in the project is publicly available and invites further education and experimentation in this research project.While the traditional methods of classification of geometric structures involve a quotient space, frequently the moduli spaces are pathological. Simple examples lead to quotients of nice spaces by chaotic group actions. The PI adopts the viewpoint that classification of geometric structures is really a dynamical system. Many well known interesting dynamical systems arise from this context. In particular the extension of the Teichmuller geodesic flow to the universal character variety over the Riemann moduli space has rich dynamics, impacting low-dimensional geometry, topology and mathematical physics. The PI will study this dynamical system as the fusion of the well-developed subject of Teichmuller dynamics and his previous work on mapping class group dynamics on character varieties and moduli spaces of geometric structures.
这个项目涉及流形上几何结构的分类。 “几何结构”是指一组坐标系,其中坐标存在于具有“经典几何”的空间中:示例包括欧几里得几何、非欧几里得几何、射影几何、仿射几何或共形几何。自世纪中期以来,对对称性的考虑强调了这样一种观点,即经典几何只是研究在李群的传递作用下不变的性质。 与此密切相关的是对称性在现代物理学中的重要性,并且本研究中的许多对象都具有物理动机。固定拓扑上的几何结构分类意味着将更严格的几何测量放在松散组织的点集合上,这就是拓扑。 一个简单的例子是球体不能支持欧几里得几何。 这只是对没有度量准确的世界地图集这一事实的数学抽象。至少有一页世界地图会扭曲距离。 另一方面,环面(甜甜圈、百吉饼或内胎的表面)确实允许欧几里得结构。事实上,把欧几里得几何放在环面上的不同方法有其自身丰富的几何学。技术在这项调查及其向公众传播方面都发挥着关键作用。许多数学家参与了这个项目,包括同事,博士后,研究生,本科生,甚至高中生。研究对象通常是二维和三维的,可以使用计算机进行可视化。在这个项目中开发的所有软件都是公开的,并邀请进一步的教育和实验在这个研究项目中。虽然传统的几何结构分类方法涉及商空间,但模空间往往是病态的。通过简单的例子,我们得到了由混沌群作用产生的好空间。PI采用的观点,分类的几何结构是一个真正的动力系统。许多著名的有趣的动力系统产生于这一背景。特别是延伸的Teichmuller测地流的普遍字符品种在黎曼模空间具有丰富的动力学,影响低维几何,拓扑和数学物理。 PI将研究这个动力系统的融合发展良好的主题Teichmuller动态和他以前的工作映射类组动态字符品种和模空间的几何结构。
项目成果
期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Tameness of Margulis space-times with parabolics
- DOI:10.1515/forum-2019-0331
- 发表时间:2017-10
- 期刊:
- 影响因子:0.8
- 作者:Suhyoung Choi;Todd A. Drumm;W. Goldman
- 通讯作者:Suhyoung Choi;Todd A. Drumm;W. Goldman
Einstein tori and crooked surfaces
爱因斯坦环面和弯曲表面
- DOI:10.1515/advgeom-2020-0023
- 发表时间:2021
- 期刊:
- 影响因子:0.5
- 作者:Burelle, Jean-Philippe;Charette, Virginie;Francoeur, Dominik;Goldman, William M.
- 通讯作者:Goldman, William M.
The mapping class group action on -character varieties
字符变体上的映射类组动作
- DOI:10.1017/etds.2020.50
- 发表时间:2021
- 期刊:
- 影响因子:0.9
- 作者:GOLDMAN, WILLIAM M.;LAWTON, SEAN;XIA, EUGENE Z.
- 通讯作者:XIA, EUGENE Z.
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William Goldman其他文献
Change in a state department of mental health: A view from within
- DOI:
10.1007/bf02253267 - 发表时间:
1976-09-01 - 期刊:
- 影响因子:2.700
- 作者:
William Goldman - 通讯作者:
William Goldman
Two steps forward and two steps back
- DOI:
10.1007/bf00706988 - 发表时间:
1993-11-01 - 期刊:
- 影响因子:2.700
- 作者:
William Goldman - 通讯作者:
William Goldman
Relating WEATHER TRENDS AND RESIDENCE LENGTH TO LOCAL CLIMATE CHANGE CONCERNS IN KNOXVILLE, TENNESSEE
将田纳西州诺克斯维尔的天气趋势和居住时间与当地气候变化问题联系起来
- DOI:
10.1080/00167428.2023.2239879 - 发表时间:
2023 - 期刊:
- 影响因子:2
- 作者:
D. Burow;Alyssa Cannistraci;Savannah A. Collins;H. Davis;William Goldman;N. Grondin;Dakotah D. Maguire;Kelsey N. Ellis - 通讯作者:
Kelsey N. Ellis
Complete Flat Affine and Lorentzian Manifolds
- DOI:
10.1023/a:1023680928912 - 发表时间:
2003-01-01 - 期刊:
- 影响因子:0.500
- 作者:
Virginie Charette;Todd Drumm;William Goldman;Maria Morrill - 通讯作者:
Maria Morrill
The ChatGPT Artificial Intelligence Chatbot: How Well Does It Answer Accounting Assessment Questions?
ChatGPT 人工智能聊天机器人:它回答会计评估问题的能力如何?
- DOI:
10.2308/issues-2023-013 - 发表时间:
2023 - 期刊:
- 影响因子:0.9
- 作者:
David A. Wood;M. P. Achhpilia;Mollie T. Adams;Sanaz Aghazadeh;Kazeem O. Akinyele;Mfon Akpan;Kristian D. Allee;Abigail Allen;E. D. Almer;Daniel Ames;Viktor Arity;Dereck Barr‐Pulliam;K. A. Basoglu;Andrew Belnap;Jeremiah W. Bentley;T. Berg;Nathan R. Berglund;Erica Berry;Avishek Bhandari;Md Nazmul Hasan Bhuyan;Paulette Black;Eva Blondeel;David Bond;Annika Bonrath;A. F. Borthick;E. S. Boyle;M. Bradford;D. M. Brandon;Joseph F. Brazel;Bryan G. Brockbank;Marcus Burger;Dmitri Byzalov;James N. Cannon;Cecile Q. Caro;Abraham H. Carr;Jack M. Cathey;Ryan Cating;K. Charron;Stacy Chavez;Jason Chen;Jennifer C. Chen;Jennifer W. Chen;Christine Cheng;Xu Cheng;Brant E. Christensen;K. Church;N. J. Cicone;Patience Constance;Lauren A. Cooper;Candice L. Correia;Joshua G. Coyne;W. Cram;Asher Curtis;Ronald J. Daigle;Steven Dannemiller;Stephan A. Davenport;Gregory S. Dawson;Karen J. De Meyst;Scott Dell;Sebahattin Demirkan;Christine A. Denison;Hrishikesh Desai;S. DeSimone;Lea Diehl;Ruth Dimes;Bei Dong;Amy M. Donnelly;Adam W Du Pon;H. Duan;Ada Duffey;R. Dunn;Mary P. Durkin;Ann C. Dzuranin;Rachel M. Eberle;Matthew S. Ege;Dina El Mahdy;Adam M. Esplin;Marc Eulerich;P. Everaert;Nusrat Farah;L. Farish;Michael Favere;Dutch Fayard;Jessica R. Filosa;Melinda Ford;Diana R. Franz;Bachman P. Fulmer;Sarah Fulmer;Z. Furner;Sonia Gantman;Steve Garner;Jace B. Garrett;Xin Geng;J. Golden;William Goldman;J. Gómez;M. Gooley;Shawn P. Granitto;Karen Y. Green;Cindy L. Greenman;Gaurav Gupta;Ronald N. Guymon;Kevin Hale;Christopher J. Harper;S. Hartt;Holly Hawk;S. Hawkins;E. M. Hawkins;D. Hay;Rafael Heinzelmann;Cassy D. Henderson;Bradley E. Hendricks;William G. Heninger;M. Hill;Nicole Holden;D. K. Holderness;Travis P. Holt;Jeffrey L. Hoopes;Sheng;Feiqi Huang;H. Huang;Ting‐Chiao Huang;Brian W. Huels;Kara Hunter;P. Hurley;Kerry K. Inger;Sharif Islam;Isaac Ison;H. Issa;Andrew B. Jackson;Scott C. Jackson;Diane J. Janvrin;Peggy D. Jimenez;Daniel Johanson;J. Judd;Brett S. Kawada;A. Kelton;Sara Kern;Jon N. Kerr;Marsha B. Keune;Mindy Kim;B. Knox;G. Kogan;Amr Kotb;Ronja Krane;Joleen Kremin;Kimberly S. Krieg;Jonathan Kugel;Ellen M. Kulset;C. Kuruppu;Garrison LaDuca;Barbara Lamberton;Melvin A. Lamboy;Bradley Lang;Stephannie A. Larocque;M. Larson;Bradley P. Lawson;James G. Lawson;Lorraine S. Lee;Margarita M. Lenk;Michelle Li;Jonathan T. Liljegren;Yi‐Hung Lin;Wu;Zishang Liu;Brandon Lock;James H. Long;Tina M. Loraas;Suzanne L. Lowensohn;Thomas R. Loy;Hakim Lyngstadaas;Wim Maas;J. MacGregor;D. Madsen;Carissa L. Malone;Maximilian Margolin;Mary E. Marshall;Rachel M. Martin;Colleen McClain Mpofu;Chris Mccoy;Nicholas McGuigan;D. McSwain;Michele D. Meckfessel;M. Mellon;Olivia S. Melton;Julie M. Mercado;Steven Mitsuda;K. Modugu;Stephen Moehrle;A. M. Chaghervand;Kevin C. Moffitt;J. Moon;Brigitte Muehlmann;John Murray;Emmanuel S. Mwaungulu;Noah Myers;J. Naegle;Martin J. Ndicu;Aaron S. Nelson;A. L. Nguyen;Thomas Niederkofler;Ehsan Nikbakht;Ann D. O'Brien;Kehinde Ogunade;Daniel E. O’Leary;M. Oler;Derek K. Oler;K. Olsen;J. I. Otalor;Kyle W. Outlaw;Michael E. Ozlanski;Jenny Parlier;Jeffrey S. Paterson;Christopher A. Pearson;M. J. Petersen;S. Petra;Matthew D. Pickard;Jeffrey S. Pickerd;R. Pinsker;Catherine Plante;James M. Plečnik;R. Price;Linda A. Quick;J. Raedy;Robyn L. Raschke;Julie Ravenscraft;Vernon Richardson;Brett A. Rixom;J. F. Robertson;Iyad Rock;Miles A. Romney;Andrea M. Rozario;Michael F. Ruff;Kathleen Rupley;A. Saeedi;Aaron Saiewitz;Leigh Salzsieder;Sayan Sarkar;Michael Saulls;Tialei A. Scanlan;Tammie J. Schaefer;Daniel Schaupp;Gary P. Schneider;A. Seebeck;R. Sellers;Samantha C. Seto;Romi;Yuxin Shan;Matthew G. Sherwood;Maggie Singorahardjo;H. Skaftadottir;Justyna Skomra;Jason L. Smith;Dallin O. Smith;James Smith;Mason C. Snow;Ryan D. Sommerfeldt;Kate B. Sorensen;Trevor L. Sorensen;Andrew C. Spieler;Matthew A. Stallings;Lesya Stallings;A. Stancill;Jonathan D. Stanley;Chad M. Stefaniak;Nathaniel M Stephens;Bryan W. Stewart;Theophanis C. Stratopoulos;Daniel A. Street;Meena Subedi;S. Summers;C. H. Sundkvist;Christina Synn;Amanuel F. Tadesse;Gregory P. Tapis;Kerri L. Tassin;Samantha Taylor;M. Teal;Ryan Teeter;M. Tharapos;Jochen C. Theis;Jack Thomas;K. Thompson;Todd A. Thornock;Wendy M. Tietz;Anthony M. Travalent;Brad S. Trinkle;J. Truelson;Michael C. Turner;Brandon Vagner;H. Vakilzadeh;Jesse van der Geest;Victor van Pelt;Scott D. Vandervelde;Jose Vega;Sandra C. Vera;Brigham Villanueva;N. Vincent;Martin Wagener;S. Walton;Rick C. Warne;Olena V. Watanabe;David Watson;M. Watson;J. Weber;T. Weirich;Ashley N. West;Amanda L. Wilford;Aaron B. Wilson;Brian Winrow;Timothy Winrow;Tasia S. Winrow;D. Wiseman;Annie L. Witte;B. D. Wood;Jessica Wood;Darryl J. Woolley;Nicole S. Wright;Juan Wu;Xiao;Dimitri Yatsenko;Courtney E. Yazzie;GL Young;C. Zhang;Aleksandra B. Zimmerman;E.N.W. Zoet - 通讯作者:
E.N.W. Zoet
William Goldman的其他文献
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{{ truncateString('William Goldman', 18)}}的其他基金
Dynamics and the Classification of Geometries on Manifolds
流形上的动力学和几何分类
- 批准号:
2203493 - 财政年份:2022
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
International Centre for Theoretical Sciences, Bangalore, India
国际理论科学中心,印度班加罗尔
- 批准号:
1261422 - 财政年份:2012
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
RNMS: Geometric Structures and Representation Varieties
RNMS:几何结构和表示种类
- 批准号:
1107367 - 财政年份:2011
- 资助金额:
$ 43.29万 - 项目类别:
Continuing Grant
FRG: COLLABORATIVE RESEARCH: DEFORMATION SPACES OF GEOMETRIC STRUCTURES
FRG:协作研究:几何结构的变形空间
- 批准号:
1065965 - 财政年份:2011
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
Advanced School and Workshops on Discrete Groups in Complex Geometry
复杂几何离散群高级学校和研讨会
- 批准号:
1010995 - 财政年份:2010
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
IMS PROGRAM ON GEOMETRY, TOPOLOGY AND DYNAMICS OF CHARACTER VARIETIES: SUMMER SCHOOL, WORKSHOP AND CONFERENCE
IMS 人物几何、拓扑和动力学项目:暑期学校、研讨会和会议
- 批准号:
0965849 - 财政年份:2010
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
CIRM Workshop: Representations of Surface Groups
CIRM 研讨会:表面基团的表示
- 批准号:
0804207 - 财政年份:2008
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
Lower and Upper Curvature Bounds: Topology vs. Geometry
曲率下界和曲率上界:拓扑与几何
- 批准号:
0604557 - 财政年份:2006
- 资助金额:
$ 43.29万 - 项目类别:
Standard Grant
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Analysis and applications of geometric Schrodinger equations: topological solitons and dynamics in ferromagnets
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Dynamics of maps with memory, random maps, multi-valued maps and the geometric Markov Renewal processes
具有记忆的映射动力学、随机映射、多值映射和几何马尔可夫更新过程
- 批准号:
RGPIN-2017-05321 - 财政年份:2022
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CAREER: Forward and Inverse Uncertainty Quantification of Cardiovascular Fluid-Structure Dynamics via Multi-fidelity Physics-Informed Bayesian Geometric Deep Learning
职业:通过多保真物理信息贝叶斯几何深度学习对心血管流体结构动力学进行正向和逆向不确定性量化
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Variational principles for continuum dynamics on geometric rough paths.
几何粗糙路径上连续介质动力学的变分原理。
- 批准号:
2478288 - 财政年份:2020
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$ 43.29万 - 项目类别:
Studentship
Analysis and applications of geometric Schrodinger equations: topological solitons and dynamics in ferromagnets
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