Fibrations and the topology of low-dimensional manifolds

纤维振动和低维流形的拓扑

基本信息

  • 批准号:
    0905380
  • 负责人:
  • 金额:
    $ 11.01万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2009
  • 资助国家:
    美国
  • 起止时间:
    2009-07-15 至 2013-06-30
  • 项目状态:
    已结题

项目摘要

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Through their connection with mapping class groups, Lefschetz fibrations and related structures on 4-manifolds provide a group-theoretic means for studying symplectic and near-symplectic 4-manifolds. This project will use Lefschetz fibrations in conjunction with invariants coming from Floer homology to study several related questions. First, the PI has shown that several geometric operations on 4-manifolds can be realized as instances of monodromy substitutions in Lefschetz fibrations, coming from new relations in the mapping class group. The PI will pursue this line of inquiry with the goal of finding new constructions of exotic 4-manifolds. Second, the PI will use the technology of relative Ozsváth-Szabó invariants (developed in joint work of the PI and S. Jabuka) to produce new examples of contact 3-manifolds admitting infinitely many topologically equivalent but smoothly distinct Stein fillings, extending previous joint work. Next is the topological meaning of the Ozsváth-Szabó invariants themselves: for example, do the Ozsváth-Szabó invariants provide constraints on the sorts of Lefschetz structures supported by a given 4-manifold? Conversely, can one calculate the Ozsváth-Szabó invariants of a Lefschetz fibration from its monodromy representation? Finally, he will continue to develop the theory of perturbed Heegaard Floer homology, which will expand the utility of the Heegaard Floer "package."Since Einstein's description of mechanics and electrodynamics as inherently a four-dimensional theory, the observed universe has generally been conceived as a smooth four-dimensional manifold: that is, a four-dimensional analog of a smooth surface such as a plane or sphere. A fundamental question is then: what manifold is it? To pose a simplified analogy, the surface of the earth is generally ``flat'' when viewed by a casual observer, but it is a mistake to infer that it is planar. The universe is similarly "mostly flat" on an appropriate distance scale, but its global topology or "shape" is not known. A goal of the 4-manifold topologist, then, is to describe the list of possibilities for the underlying structure of the universe, in analogy with the relatively easy-to-understand list of possible surfaces from which a "generally flat" object like the surface of the earth can select (sphere, torus, etc.). Perhaps not coincidentally, the theory of smooth 4-dimensional manifolds is vastly more complicated than the analogous theory in any other dimension. Indeed, surprisingly little is known regarding important and basic existence and uniqueness questions for smooth 4-manifolds. The work supported by this grant will approach several of these questions by making use of a Lefschetz fibration (or similar geometric structure) on a 4-manifold, together with the invariants of 3- and 4-dimensional manifolds provided by recently-developed mathematical tools that are based on ideas from gauge-theoretic physics.
该奖项由2009年美国复苏与再投资法(公法111-5)资助,通过它们与映射类群的联系,4-流形上的Lefschetz颤动及其相关结构为研究辛和近辛4-流形提供了一种群论方法。这个项目将使用Lefschetz纤颤结合来自Floer同调的不变量来研究几个相关的问题。首先,PI证明了4-流形上的几种几何运算可以由映射类群中的新关系实现为Lefschetz原函数中的单行替换的实例。PI将继续这条线的调查,目标是寻找奇异的4-流形的新结构。其次,PI将使用相对Ozsváth-Szabó不变量技术(在PI和S.Jabuka的联合工作中开发)来产生接触3-流形的新示例,允许无限多个拓扑等价但光滑不同的Stein填充,扩展了以前的联合工作。其次是Ozsváth-Szabó不变量本身的拓扑意义:例如,Ozsváth-Szabó不变量是否对给定的4-流形所支持的Lefschetz结构的种类提供约束?相反,我们能从Lefschetz纤维的单列表示中计算出它的Ozsváth-Szabó不变量吗?最后,他将继续发展扰动Heegaard Floer同调理论,这将扩展Heegaard Floer“包”的用途。由于爱因斯坦将力学和电动力学描述为固有的四维理论,因此观察到的宇宙通常被设想为光滑的四维流形:即平面或球面等光滑表面的四维模拟。那么,一个根本的问题是:它是什么流形?打个简单的比方说,一般人看地球表面是“平的”,但推论它是平的是错误的。同样,宇宙在适当的距离尺度上是“基本平坦的”,但它的全局拓扑或“形状”是未知的。因此,4-流形拓扑学家的目标是描述宇宙潜在结构的可能性列表,类似于相对容易理解的可能表面列表,像地球表面这样的“一般平坦”的物体可以从中选择(球体、环面等)。也许并非巧合的是,光滑的四维流形理论比其他任何维度的类似理论都要复杂得多。事实上,令人惊讶的是,关于光滑4-流形的重要的和基本的存在唯一性问题知之甚少。这项由这笔拨款支持的工作将通过利用4维流形上的Lefschetz纤维(或类似的几何结构)以及最近开发的数学工具提供的3维和4维流形的不变量来解决其中的几个问题,这些工具基于规范理论物理学的思想。

项目成果

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Thomas Mark其他文献

EFFECTS OF SEMAGLUTIDE VERSUS COMPARATORS ON CARDIOVASCULAR EVENTS ACROSS A CONTINUUM OF BASELINE CARDIOVASCULAR RISK: COMBINED ANALYSIS OF THE SUSTAIN AND PIONEER TRIALS
  • DOI:
    10.1016/s0735-1097(20)32536-5
  • 发表时间:
    2020-03-24
  • 期刊:
  • 影响因子:
  • 作者:
    Mansoor Husain;Stephen C. Bain;Anders G. Holst;Thomas Mark;Søren Rasmussen;Ildiko Lingvay
  • 通讯作者:
    Ildiko Lingvay
: Full list
:完整列表
  • DOI:
  • 发表时间:
    2017
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Thomas Mark;Hutchcroft;Lisa Fong;Jonathan Hermon
  • 通讯作者:
    Jonathan Hermon
17 - Cardiovascular Safety and Severe Hypoglycemia Benefit of Insulin Degludec vs. Insulin Glargine U100 in Older Patients (≥65 Years) with Type 2 Diabetes: Observations From DEVOTE
  • DOI:
    10.1016/j.jcjd.2018.08.020
  • 发表时间:
    2018-10-01
  • 期刊:
  • 影响因子:
  • 作者:
    Richard E. Pratley;Scott S. Emerson;Edward Franek;Matthew P. Gilbert;Steven P. Marso;Darren K. McGuire;Thomas R. Pieber;Neil R. Poulter;Charlotte T. Hansen;Melissa V. Hansen;Thomas Mark;Alan C. Moses;Bernard Zinman;Jina Hahn
  • 通讯作者:
    Jina Hahn

Thomas Mark的其他文献

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{{ truncateString('Thomas Mark', 18)}}的其他基金

RTG: Geometry and Topology at the University of Virginia
RTG:弗吉尼亚大学的几何和拓扑
  • 批准号:
    1839968
  • 财政年份:
    2019
  • 资助金额:
    $ 11.01万
  • 项目类别:
    Continuing Grant
Virginia Topology Conference 2018
2018 年弗吉尼亚拓扑会议
  • 批准号:
    1839925
  • 财政年份:
    2018
  • 资助金额:
    $ 11.01万
  • 项目类别:
    Standard Grant
Low-Dimensional Contact and Symplectic Topology
低维接触和辛拓扑
  • 批准号:
    1309212
  • 财政年份:
    2013
  • 资助金额:
    $ 11.01万
  • 项目类别:
    Standard Grant
Conference in Honor of Ronald Fintushel
纪念罗纳德·芬图谢尔的会议
  • 批准号:
    0506737
  • 财政年份:
    2005
  • 资助金额:
    $ 11.01万
  • 项目类别:
    Standard Grant

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Fibered纽结的自同胚、Floer同调与4维亏格
  • 批准号:
    12301086
  • 批准年份:
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  • 资助金额:
    30.00 万元
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Domain理论与拓扑学研究
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    2004
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    7.0 万元
  • 项目类别:
    面上项目

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Conference: Combinatorial and Analytical methods in low-dimensional topology
会议:低维拓扑中的组合和分析方法
  • 批准号:
    2349401
  • 财政年份:
    2024
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    $ 11.01万
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    Standard Grant
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    EP/Y004256/1
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    2024
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Re-examination of classical problems in low-dimensional topology from higher invariants
从更高的不变量重新审视低维拓扑中的经典问题
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CAREER: Low dimensional topology via Floer theory
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