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Pseudoholomorphic curves in low-dimensional topology

Pseudoholomorphic curves in low-dimensional topology
低维拓扑中的伪全纯曲线
批准号:
0806037
负责人:
Michael Hutchings
金额:
$34.16万
依托单位国家:
美国
项目类别:
Standard Grant
财政年份:
2008
资助国家:
美国
项目状态:
已结题
起止时间:
2008-07-01 至 2012-06-30

项目摘要

项目成果

Michael Hutchings的其他基金

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中文摘要
翻译
摘要奖:DMS-0806037首席研究员:Michael L.Hutchings该项目的主要部分是开发一种新的接触三流形的不变量,它是根据Reeb流的周期轨道和辛化中的嵌入伪全纯曲线定义的。ECH猜想同构于Seiberg-Witten和Ozsvath-Szabo Floer同调的版本。该项目的一些具体目标是进一步发展ECH的分析基础;创建计算ECH的工具,特别是在Openbook分解方面;使用ECH获得Reeb轨道数的下界;以及致力于将ECH扩展为更一般的理论,将其与Ozsvath-Szabo Floer同调统一。一些更广泛的目标是使用ECH机制来帮助计算三维辛场理论,并探索族的Floer理论不变量。本项目中开发的嵌入接触同调位于动力学和低维拓扑之间的界面上。动力学涉及物理系统的超时行为,而低维拓扑研究三维和四维弯曲空间的可能形状。嵌入式接触学允许人们从对动力学的理解中获得深刻的拓扑信息,反之,从拓扑条件中获得重要的动力学信息,如稳定构型的存在。
英文摘要
AbstractAward: DMS-0806037Principal Investigator: Michael L. HutchingsThe main part of the project is to develop "embedded contacthomology" (ECH), a new invariant of a contact three-manifold,which is defined in terms of periodic orbits of the Reeb flow andembedded pseudoholomorphic curves in the symplectization. ECH isconjecturally isomorphic to versions of the Seiberg-Witten andOzsvath-Szabo Floer homologies. Some specific goals of theproject are to further develop the analytical foundations of ECH;to create tools for computing ECH, particularly in terms of openbook decompositions; to use ECH to obtain lower bounds on numbersof Reeb orbits; and to work towards extending ECH to a moregeneral theory which would unify it with the Ozsvath-Szabo Floerhomology. Some broader goals are to use ECH machinery to helpcompute symplectic field theory in three dimensions, and toexplore Floer-theoretic invariants of families.The embedded contact homology developed in this project lies onthe interface between dynamics and low-dimensional topology.Dynamics is concerned with the behavior of physical systems overtime, while low-dimensional topology studies the possible shapesof curved spaces in three and four dimensions. Embedded contacthomology allows one to obtain deep topological information froman understanding of dynamics; and conversely to obtain importantdynamical information, such as the existence of stableconfigurations, from topological conditions.
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Contact homology, dynamics, and embeddings
  • 批准号:
    2005437
  • 项目类别:
    Standard Grant
  • 资助金额:
    $23.97万
  • 财政年份:
    2020
  • 负责人:
    Michael Hutchings
  • 依托单位:
Current Trends in Symplectic Topology
  • 批准号:
    1916934
  • 项目类别:
    Standard Grant
  • 资助金额:
    $1.5万
  • 财政年份:
    2019
  • 负责人:
    Michael Hutchings
  • 依托单位:
Contact Homology and Quantitative Symplectic Geometry
  • 批准号:
    1708899
  • 项目类别:
    Continuing Grant
  • 资助金额:
    $33.0万
  • 财政年份:
    2017
  • 负责人:
    Michael Hutchings
  • 依托单位:
The dynamics of antimicrobial resistance gene prevalence on a commercial pig farm: implications for policy
  • 批准号:
    NE/N019806/1
  • 项目类别:
    Research Grant
  • 资助金额:
    $7.73万
  • 财政年份:
    2016
  • 负责人:
    Michael Hutchings
  • 依托单位:
国内基金
海外基金
Lienard系统的不变代数曲线、可积性与极限环问题研究
  • 批准号:
    12301200
  • 项目类别:
    青年科学基金项目
  • 资助金额:
    30.00万元
  • 批准年份:
    2023
  • 负责人:
    钱欣洁
  • 依托单位: