CAREER: Floer theories and Reeb dynamics of contact manifolds

职业:Floer 理论和接触流形的 Reeb 动力学

基本信息

  • 批准号:
    2142694
  • 负责人:
  • 金额:
    $ 43.57万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2022
  • 资助国家:
    美国
  • 起止时间:
    2022-09-01 至 2027-08-31
  • 项目状态:
    未结题

项目摘要

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The mathematical structures known as contact and symplectic manifolds have their origins in the study of classical mechanical systems from physics, which allow one to describe systems such as planetary motion and wave propagation. The equations of motion can be described in terms of mathematical objects known as flow lines of Hamiltonian and Reeb vector fields. Understanding the dynamics of these vector fields led to the development of global Floer theoretic invariants of symplectic and contact manifolds. By developing foundations and applications of these invariants, the PI's work will illuminate the interconnectedness of dynamics, embeddings, knot theory, geometry, and topology. The PI will expand her efforts to increase the access and success of underrepresented students in mathematics and academia more broadly. She is conducting, jointly with others, two national studies to delineate forms of antiracism in academic advising in STEM fields and to examine the effect of academic advisors' practices on BIPOC (Black, Indigenous, and people of color) PhD student psychological experiences and outcomes. Based on the results of the studies, she will construct a set of best practices and effective behaviors in academic advising. To train future generations, the PI will co-organize two international conferences with professional development programming for early-career mathematicians, including a panel discussion on academic jobs for graduate students. She will also develop and implement summer research programs for undergraduates that will include graduate students and a postdoctoral researcher.The project concerns Floer theoretic invariants and Reeb dynamics of contact manifolds. The dynamics of Reeb vector fields are subtly related to the underlying contact structure as well as the topology and geometry of the underlying manifold. Reeb vector fields realize distance minimizing flow and arise as the restriction of Hamiltonian vector fields to contact type hypersurfaces and boundaries of symplectic manifolds. Reeb dynamics additionally govern embeddings between symplectic manifolds with contact type boundary as well as the topology of symplectic fillings of contact manifolds. The PI's research has a particular emphasis on the study of closed periodic Reeb orbits (circular flow lines) by way of the development of various Floer theories, so as to capture different dynamical, geometric, and topological phenomenon. She will also provide applications to surface dynamics and the study of contact and symplectic manifolds. The project includes community building, mentorship, recruitment, and retention programming for undergraduate and graduate students in mathematics, as well as the training of students in communication skills.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。被称为接触流形和辛流形的数学结构起源于从物理学对经典机械系统的研究,这使得人们能够描述行星运动和波传播等系统。运动方程可以用称为哈密顿矢量场和里布矢量场流线的数学对象来描述。理解这些矢量场的动力学导致了辛流形和接触流形的整体Floer理论不变量的发展。通过发展这些不变量的基础和应用,PI的工作将阐明动力学、嵌入、纽结理论、几何和拓扑的相互联系。国际学生联合会将扩大她的努力,以增加在更广泛的数学和学术界中代表性不足的学生的机会和成功。她正在与其他人共同开展两项全国性研究,以描述STEM领域学术咨询中反种族主义的形式,并检查学术顾问的做法对BIPOC(黑人、土著和有色人种)博士生心理体验和结果的影响。根据研究结果,她将构建一套最佳实践和有效的学术建议行为。为了培养下一代,国际数学家协会将联合举办两个针对初出茅庐的数学家的专业发展规划的国际会议,包括关于研究生学术工作的小组讨论。她还将为本科生制定和实施暑期研究计划,其中将包括研究生和博士后研究人员。该项目涉及Floer理论不变量和接触流形的Reeb动力学。Reeb矢量场的动力学与底层接触结构以及底层流形的拓扑和几何有微妙的关系。Reeb矢量场实现距离最小化流动,是由于哈密顿矢量场对接触型超曲面和辛流形边界的限制而产生的。此外,Reeb动力学还控制了具有接触型边界的辛流形之间的嵌入以及接触流形的辛填充的拓扑。PI的研究侧重于通过发展各种Floer理论来研究闭合周期Reeb轨道(圆形流线),以捕捉不同的动力学、几何和拓扑现象。她还将提供曲面动力学的应用以及接触流形和辛流形的研究。该项目包括为本科生和研究生建立社区、指导、招聘和保留规划,以及培训学生的沟通技能。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

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Joanna Nelson其他文献

269 – Naegleria fowleri
269 – 福氏耐格里阿米巴
  • DOI:
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Joanna Nelson;Upinder Singh
  • 通讯作者:
    Upinder Singh

Joanna Nelson的其他文献

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{{ truncateString('Joanna Nelson', 18)}}的其他基金

Pseudoholomorphic Invariants of Contact Manifolds
接触流形的伪全纯不变量
  • 批准号:
    2104411
  • 财政年份:
    2021
  • 资助金额:
    $ 43.57万
  • 项目类别:
    Standard Grant
Moduli Problems in Contact Geometry
接触几何中的模量问题
  • 批准号:
    1840723
  • 财政年份:
    2018
  • 资助金额:
    $ 43.57万
  • 项目类别:
    Continuing Grant
Moduli Problems in Contact Geometry
接触几何中的模量问题
  • 批准号:
    1810692
  • 财政年份:
    2018
  • 资助金额:
    $ 43.57万
  • 项目类别:
    Continuing Grant
PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    1303903
  • 财政年份:
    2013
  • 资助金额:
    $ 43.57万
  • 项目类别:
    Fellowship Award

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Fibered纽结的自同胚、Floer同调与4维亏格
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瞬子Floer同调与Khovanov同调
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三维切触拓扑,Heegaard Floer同调,和范畴化
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    11601256
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    2016
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    19.0 万元
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    青年科学基金项目
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    11526115
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    2015
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    数学天元基金项目
三维流形的Floer同调
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    11001147
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    2010
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Floer Theories for 3-Manifolds
3 流形的 Floer 理论
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Higher Structure in Low-Dimensional Floer Theories
低维Floor理论中的高级结构
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    1810893
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    $ 43.57万
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    2017
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Contact structures, open books, and connections between Heegaard Floer homology and the Khovanov-Rozansky link homology theories
Heegaard Floer 同调与 Khovanov-Rozansky 链接同调理论之间的联系结构、开放书籍以及联系
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    1251064
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    2012
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    $ 43.57万
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Contact structures and Floer homology theories
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    1105432
  • 财政年份:
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Heegaard Floer 同调与 Khovanov-Rozansky 链接同调理论之间的联系结构、开放书籍以及联系
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Khovanov 和 Heegaard Floer 型同调理论之间的联系
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辛几何和低维拓扑中的弗洛尔理论
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