Topology of non-positively curved manifolds

非正曲流形的拓扑

基本信息

  • 批准号:
    1405185
  • 负责人:
  • 金额:
    $ 12.81万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2014
  • 资助国家:
    美国
  • 起止时间:
    2014-08-01 至 2017-07-31
  • 项目状态:
    已结题

项目摘要

A fundamental question of Geometry and Topology is how the global topology of a space is determined by local geometric data, in particular, the curvature of the space. Nonpositively curved manifolds (spaces) has been a subject where there are a lot of rich theorems, and it has been a central object of study not only in geometry, but also in related fields of mathematics, such as geometric group theory and dynamics. The field, however, seems to have been abandoned in the last two decades although there are a lot of unanswered questions, some of which are within reach. Constructions of smooth, nonpositively curved manifolds are poorly understood in the sense that most examples known are of a very restrictive type, such as hyperbolic manifolds and pinched negatively curved manifolds, which do not illustrate more generic properties of nonpositively curved spaces. One of the aims of the proposed research is to construct more nonpositively curved manifolds, especially with interesting properties that are not seen in hyperbolic or pinched negatively curved manifolds. If a space is known to have a nonpositively curved metric, one can deduce a lot about it. The question whether a space admits a nonpositively curved metric has become of great interest in different areas in topology. The proposed research also aims to find topological properties shared by large classes of nonpositively curved manifolds and topological obstructions to having a nonpositively curved metric. Specifically, the PI will study the topology of ends as obstructions to having nonpositively curved metrics. The project aims to study noncompact, complete, finite volume, bounded, nonpositively curved manifolds M. Objects to be studied are the topology of ends, invariants of the fundamental groups of these manifolds (such as the cohomological dimension and the action dimension), and the relation between the topology of the end of such a manifold of the set of non-horospherical limit points. The problem of how to distinguish which locally CAT(0) manifolds have a smooth nonpositively curved Riemannian metric will also be studied. In particular, in the case when M has tame ends, the question which manifold C can occur as the cross section of each end of M will be studied extensively. In low dimension, such as when M has dimension 4, the goal is to show that each cross section of an end of M is aspherical by computing the second homotopy group of the ends of M. Once this has been proven, and with the result of the Geometrization Theorem in three-manifold theory, a classification of all manifolds C that can occur as cross sections may be within reach, or at least progress can be made using recent techniques of Ontaneda's Riemannian hyperbolization procedure. The relation between the topology of the end of such a manifold and the set of non-horospherical limit points will be studied. For the case when M is a locally symmetric space of noncompact type, it will be shown that the set of non-horospherical limit points is the same as the rational Tits building using Saper's tilings. The question whether a similar phenomenon happens in more general nonpositively curved settings will be investigated. This project also aims to construct new examples of locally CAT(0) manifolds that do not admit a smooth nonpositively curved metric. Obstructions that will be used to distinguish these from smooth nonpositively curved manifolds are invariants at infinity that come from embedded flat tori in the manifolds that are locally linked with linking number greater than one.
几何与拓扑学的一个基本问题是空间的全局拓扑如何由局部几何数据,特别是空间的曲率决定。非正曲流形(空间)是一个有着丰富定理的学科,它不仅是几何学的中心研究对象,也是相关数学领域的中心研究对象,如几何群论和动力学。 然而,在过去的二十年里,这个领域似乎已经被遗弃了,尽管有很多问题没有得到回答,其中一些问题已经触手可及。光滑的非正曲流形的构造很少被理解,因为大多数已知的例子都是非常严格的类型,例如双曲流形和收缩的负曲流形,它们不能说明非正曲空间的更一般的性质。所提出的研究的目的之一是构造更多的非正弯曲流形,特别是在双曲或捏负弯曲流形中看不到的有趣的性质。如果已知一个空间有一个非正曲度量,人们可以推导出很多关于它的问题。一个空间是否允许一个非正曲度量的问题已经成为拓扑学不同领域的极大兴趣。拟议的研究还旨在找到大类非正弯曲流形和拓扑障碍,有一个非正弯曲度量共享的拓扑性质。具体而言,PI将研究末端的拓扑结构,作为具有非正弯曲度量的障碍。 该项目旨在研究非紧的、完备的、有限体积的、有界的、非正曲流形M。研究的对象是端点的拓扑,这些流形的基本群的不变量(如上同调维数和作用维数),以及这种流形的端点的拓扑与非时球极限点集的拓扑之间的关系。如何区分哪些局部CAT(0)流形具有光滑的非正曲黎曼度量的问题也将被研究。特别地,当M具有驯服的端点时,哪个流形C可以作为M的每个端点的横截面出现的问题将被广泛地研究。在低维情况下,例如当M的维数为4时,目标是通过计算M的端部的第二同伦群来证明M的端部的每个横截面是非球面的。一旦证明了这一点,并且有了三流形理论中的几何化定理的结果,所有可以作为横截面出现的流形C的分类可能是触手可及的,或者至少可以使用Ontaneda的黎曼双曲化过程的最新技术取得进展。研究了这类流形的端点拓扑与非次球面极限点集之间的关系。当M是一个局部对称的非紧型空间时,我们证明了非准球面极限点的集合与用Saper的平铺构造的有理Tits是相同的。问题是否类似的现象发生在更一般的非正曲线设置将进行调查。该项目还旨在构建不允许光滑非正弯曲度量的局部CAT(0)流形的新例子。将被用来区分这些光滑非正弯曲流形的障碍是无穷远处的不变量,这些不变量来自流形中的嵌入平坦环面,这些流形局部地与大于1的链接数相关联。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Tu Tam Nguyen Phan其他文献

Tu Tam Nguyen Phan的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

相似国自然基金

基于深穿透拉曼光谱的安全光照剂量的深层病灶无创检测与深度预测
  • 批准号:
    82372016
  • 批准年份:
    2023
  • 资助金额:
    48.00 万元
  • 项目类别:
    面上项目
Non-CG DNA甲基化平衡大豆产量和SMV抗性的分子机制
  • 批准号:
    32301796
  • 批准年份:
    2023
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目
G蛋白偶联受体GPR110调控Lp-PLA2抑制非酒精性脂肪性肝炎的作用及机制研究
  • 批准号:
    82370865
  • 批准年份:
    2023
  • 资助金额:
    49.00 万元
  • 项目类别:
    面上项目
long non-coding RNA(lncRNA)-activatedby TGF-β(lncRNA-ATB)通过成纤维细胞影响糖尿病创面愈合的机制研究
  • 批准号:
    LQ23H150003
  • 批准年份:
    2023
  • 资助金额:
    0.0 万元
  • 项目类别:
    省市级项目
染色体不稳定性调控肺癌non-shedding状态及其生物学意义探索研究
  • 批准号:
    82303936
  • 批准年份:
    2023
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目
犬尿氨酸酶KYNU参与非酒精性脂肪肝进展为肝纤维化的作用和机制研究
  • 批准号:
    82370874
  • 批准年份:
    2023
  • 资助金额:
    49.00 万元
  • 项目类别:
    面上项目
变分法在双临界Hénon方程和障碍系统中的应用
  • 批准号:
    12301258
  • 批准年份:
    2023
  • 资助金额:
    30.00 万元
  • 项目类别:
    青年科学基金项目
BTK抑制剂下调IL-17分泌增强CD20mb对Non-GCB型弥漫大B细胞淋巴瘤敏感性
  • 批准号:
    n/a
  • 批准年份:
    2022
  • 资助金额:
    10.0 万元
  • 项目类别:
    省市级项目
Non-TAL效应子NUDX4通过Nudix水解酶活性调控水稻白叶枯病菌致病性的分子机制
  • 批准号:
  • 批准年份:
    2022
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目
一种新non-Gal抗原CYP3A29的鉴定及其在猪-猕猴异种肾移植体液排斥反应中的作用
  • 批准号:
  • 批准年份:
    2022
  • 资助金额:
    33 万元
  • 项目类别:
    地区科学基金项目

相似海外基金

Boundary representations of non-positively curved groups
非正弯曲群的边界表示
  • 批准号:
    EP/V002899/1
  • 财政年份:
    2021
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Research Grant
Apathy in Alzheimer's Disease: Investigation of the Interaction between Proline and COMT for Treatment Targeting to Positively Impact Quality of Life
阿尔茨海默氏病的冷漠:研究脯氨酸和 COMT 之间的相互作用,以积极影响生活质量为目标的治疗
  • 批准号:
    9761938
  • 财政年份:
    2018
  • 资助金额:
    $ 12.81万
  • 项目类别:
The geometry of non-positively curved groups
非正曲群的几何形状
  • 批准号:
    1812061
  • 财政年份:
    2018
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Standard Grant
Residual properties of non-positively curved groups
非正曲群的残差性质
  • 批准号:
    1992790
  • 财政年份:
    2017
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Studentship
Analytic L2-invariants of non-positively curved spaces
非正弯曲空间的解析 L2 不变量
  • 批准号:
    338540207
  • 财政年份:
    2017
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Priority Programmes
Aspects of Non-positively Curved Groups
非正曲群的方面
  • 批准号:
    418456-2012
  • 财政年份:
    2017
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Discovery Grants Program - Individual
Fixed point properties on Busemann Non-positively curved spaces, expanders, and generalizations
Busemann 上的不动点属性 非正弯曲空间、扩展器和泛化
  • 批准号:
    17H04822
  • 财政年份:
    2017
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Grant-in-Aid for Young Scientists (A)
Aspects of Non-positively Curved Groups
非正曲群的方面
  • 批准号:
    418456-2012
  • 财政年份:
    2015
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Discovery Grants Program - Individual
coarse geometry of negatively curved spaces beyond relatively hyperbolic groups
相对双曲群之外的负弯曲空间的粗略几何
  • 批准号:
    15K17528
  • 财政年份:
    2015
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Grant-in-Aid for Young Scientists (B)
Nonlinear spectral gap with respect to non-positively curved spaces
相对于非正弯曲空间的非线性谱间隙
  • 批准号:
    15K17538
  • 财政年份:
    2015
  • 资助金额:
    $ 12.81万
  • 项目类别:
    Grant-in-Aid for Young Scientists (B)
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了