CAREER: Hyperbolic geometry and knots and links

职业:双曲几何以及结和链接

基本信息

  • 批准号:
    1252687
  • 负责人:
  • 金额:
    $ 37.32万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2013
  • 资助国家:
    美国
  • 起止时间:
    2013-06-01 至 2016-09-30
  • 项目状态:
    已结题

项目摘要

It has been known since the early 1980s that knot and link complements decompose into pieces admitting a geometric structure, with the most common geometry being hyperbolic. However, the connection between hyperbolic geometry and other knot and link invariants is still not well understood. The investigator will use recent developments and techniques in 3-manifold geometry and topology to make connections between hyperbolic geometry and link invariants, focusing on problems in two areas. First, she will relate hyperbolic geometry to quantum invariants, in particular continuing her recent work to find connections between hyperbolic geometry and the colored Jones polynomial. Second, she will obtain bounds on hyperbolic quantities based on diagrammatical and topological invariants of knots and links, for example bounding volume and cusp volume, and finding isotopy classes of geodesics. As part of the educational portion of this work, she will organize two conferences on connections of hyperbolic geometry, and continue her work with undergraduate, graduate, and K-12 students.This project concerns the study of 3-dimensional spaces called 3-manifolds, which include the space of our universe (with three spatial dimensions). These spaces appear in physics, mechanics, microbiology, and chemistry, and so we wish to better understand their mathematical properties. One way to study 3-manifolds is to drill out tubes around circles from a 3-dimensional sphere, and then reattach the tubes in a different manner. The space of drilled tubes about circles is called a link complement, or knot complement if there is just one circle. In this way, knot and link complements are building blocks for 3-manifolds. The investigator will study the geometry of knot and link complements, with the hope of finding new results on the properties of broader classes of 3-manifolds. This project includes many provisions for training students. Much of the research will be carried out with the assistance of undergraduate and graduate students. In addition, as part of this project the investigator will run two research conferences, during which several graduate students and postdoctoral researchers will be invited to present their related work. Finally, the investigator will continue to run mathematical workshops for children throughout the school year.
自20世纪80年代初以来,人们就知道纽结和链环补集分解成允许几何结构的片段,其中最常见的几何结构是双曲的。 然而,双曲几何与其他纽结和链环不变量之间的联系仍然没有得到很好的理解。研究人员将使用最新的发展和技术在3流形几何和拓扑,使双曲几何和链接不变量之间的连接,集中在两个领域的问题。 首先,她将双曲几何与量子不变量联系起来,特别是继续她最近的工作,寻找双曲几何与有色琼斯多项式之间的联系。 其次,她将获得基于结和链接的数学和拓扑不变量的双曲量的界限,例如包围体积和尖点体积,并找到测地线的合痕类。 作为这项工作的教育部分的一部分,她将组织两个会议的双曲几何的连接,并继续她的工作与本科生,研究生,和K-12学生.这个项目涉及的三维空间称为三维流形,其中包括我们的宇宙空间(三维空间)的研究. 这些空间出现在物理学、力学、微生物学和化学中,因此我们希望更好地理解它们的数学性质。 研究三维流形的一种方法是从三维球体中钻取围绕圆的管,然后以不同的方式重新连接管。 钻孔管关于圆的空间称为环补,如果只有一个圆,则称为纽结补。 这样,纽结补和链补是三维流形的构建块。 研究人员将研究结和链补的几何,希望找到更广泛的3-流形类的性质的新结果。 该项目包括许多培训学生的规定。 大部分研究将在本科生和研究生的协助下进行。 此外,作为该项目的一部分,研究人员将举办两次研究会议,期间将邀请几名研究生和博士后研究人员介绍他们的相关工作。 最后,研究员将继续在整个学年为儿童举办数学讲习班。

项目成果

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Jessica Purcell其他文献

Are societies resilient? Challenges faced by social insects in a changing world
  • DOI:
    10.1007/s00040-018-0663-2
  • 发表时间:
    2018-10-01
  • 期刊:
  • 影响因子:
    1.500
  • 作者:
    Kaleigh Fisher;Mari West;Adriana M. Lomeli;S. Hollis Woodard;Jessica Purcell
  • 通讯作者:
    Jessica Purcell
Supergenes in organismal and social development of insects: ideas and opportunities
  • DOI:
    10.1016/j.cois.2024.101303
  • 发表时间:
    2025-04-01
  • 期刊:
  • 影响因子:
  • 作者:
    Jessica Purcell;Alan Brelsford
  • 通讯作者:
    Alan Brelsford

Jessica Purcell的其他文献

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{{ truncateString('Jessica Purcell', 18)}}的其他基金

CAREER: Integrating genetic and ecological drivers of a social phenotype: dynamics of a social polymorphism and supergene
职业:整合社会表型的遗传和生态驱动因素:社会多态性和超基因的动态
  • 批准号:
    1942252
  • 财政年份:
    2020
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Continuing Grant
SG: Understanding the genetic and behavioral basis of novel social phenotypes in damaging invasive wasps
SG:了解破坏性入侵黄蜂的新型社会表型的遗传和行为基础
  • 批准号:
    1655963
  • 财政年份:
    2017
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
Moab Topology Conference 2012
2012 年 Moab 拓扑会议
  • 批准号:
    1202922
  • 财政年份:
    2012
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
Collaborative research: Hyperbolic geometry of knots and 3-manifolds
合作研究:结和三流形的双曲几何
  • 批准号:
    1007437
  • 财政年份:
    2010
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
Moab Topology Conference; Moab, UT; May 2009
摩押拓扑会议;
  • 批准号:
    0932037
  • 财政年份:
    2009
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
Geometry and Topology of Knots and Links
结和链接的几何和拓扑
  • 批准号:
    0704359
  • 财政年份:
    2007
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
GRADUATE RESEARCH FELLOWSHIPS
研究生研究奖学金
  • 批准号:
    0543087
  • 财政年份:
    2005
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Fellowship Award

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Hyperbolic Geometry and Gravitational Waves
双曲几何和引力波
  • 批准号:
    2309084
  • 财政年份:
    2023
  • 资助金额:
    $ 37.32万
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Topics in Non-Euclidean / Hyperbolic Geometry
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    2890480
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Arithmetic Applications of Definable and Hyperbolic Geometry
可定义几何和双曲几何的算术应用
  • 批准号:
    RGPIN-2019-04178
  • 财政年份:
    2022
  • 资助金额:
    $ 37.32万
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    Discovery Grants Program - Individual
The Jones Polynomial and Hyperbolic Geometry of Surfaces
曲面的琼斯多项式和双曲几何
  • 批准号:
    2203255
  • 财政年份:
    2022
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    $ 37.32万
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The Geometry of Hyperbolic 3-Manifolds
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  • 批准号:
    2202718
  • 财政年份:
    2022
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    $ 37.32万
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Hyperbolic Geometry and Quantum Invariants
双曲几何和量子不变量
  • 批准号:
    2203334
  • 财政年份:
    2022
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    $ 37.32万
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Geometry and dynamics of systems with a hyperbolic flavor
双曲线系统的几何和动力学
  • 批准号:
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  • 财政年份:
    2022
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Discovery Grants Program - Individual
Hyperbolic manifolds from a contact geometry perspective
从接触几何角度看双曲流形
  • 批准号:
    2750796
  • 财政年份:
    2022
  • 资助金额:
    $ 37.32万
  • 项目类别:
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Towards a mathematical description of complex networks: Effective structure and latent hyperbolic geometry
走向复杂网络的数学描述:有效结构和潜在双曲几何
  • 批准号:
    RGPIN-2019-05183
  • 财政年份:
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  • 资助金额:
    $ 37.32万
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    Discovery Grants Program - Individual
Conference on Complex Hyperbolic Geometry and Related Topics
复杂双曲几何及相关主题会议
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    2225583
  • 财政年份:
    2022
  • 资助金额:
    $ 37.32万
  • 项目类别:
    Standard Grant
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