Measure on the Ideal Boundary of a Nonpositively Curved Space: Random Walks and Rigidity
非正弯曲空间理想边界的测量:随机游走和刚度
基本信息
- 批准号:0608643
- 负责人:
- 金额:$ 12.36万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2006
- 资助国家:美国
- 起止时间:2006-07-01 至 2011-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The combination of differential geometric and dynamical methods has been very successful in the study of negatively curved manifolds and metric spaces. For instance, delicate information about geodesics and geometry at different scales in these spaces can be gleaned from the behavior of random walks on discrete models and the study of certain ergodic measures on their geometric boundaries. For random walks on nonamenable groups, a basic question has been to understand the relationship of its Poisson boundary to other natural geometric boundaries. Starting with the work of Furstenberg, and continuing with the work of many others, much progress has been made in understanding when Poisson boundary measures for a random walks on important classes of nonpositively curved groups can be supported on their geodesic boundary. However, much less is known about what measures can arise this way. The first part of the proposed research seeks to show that many of the classes of ergodic measures arising from geometric constructions on the ideal boundary are represented by Poisson boundaries. We are also interested in groups which are not nonpositively curved, yet share some common features such as the mapping class groups or the diffeomorphism group of a circle. This represents a natural outgrowth of the PI's work with R. Muchnik. The second proposed direction of study examines the barycenter method as a tool for understanding manifolds admitting nontrivial maps to nonpositively curved manifolds. This is a differential geometric application of the study of boundary measures. By relating the volume and large scale geometry of a manifold, we wish to use these methods to realize further extensions of Mostow rigidity. A number of remarkable developments in both mathematics and the physical sciences have revealed how random processes in a given system often reflect certain structural features of that system. For example, an ant randomly stepping one unit north, south, east or west in the Euclidean plane will eventually return to its starting point with probabilistic certainty. However, this no longer holds when one allows an additional degree of freedom of movement, say up and down in the third dimension. Hence, the recurrence property of this "random walk" detects the dimension of the ambient space. We propose to study the flexibility of such connections between the geometry of the underlying space and certain random processes. We especially are interested in understanding when generalized random walks on important families of spaces can produce a prescribed set of measurements. From another point of view, these auxiliary measurements themselves capture other intrinsic aspects of these spaces, and can sometimes indicate "rigidity" of the space. This refers to the phenomenon whereby a weak equivalence between spaces implies a strong equivalence. We hope to discover new ways in which rigidity arises.
微分几何方法和动力学方法的结合在负曲流形和度量空间的研究中已经非常成功。例如,从离散模型上随机游动的行为和对几何边界上某些遍历测度的研究中,可以收集到关于这些空间中不同尺度下测地线和几何的精细信息。对于非顺从群上的随机游动,一个基本问题是理解它的泊松边界与其他自然几何边界的关系。从工作的Furstenberg,并继续与许多其他人的工作,取得了很大进展,了解当泊松边界措施的随机游动的重要类的非正曲组可以支持其测地线边界。然而,人们对这种方式可以产生什么样的措施知之甚少。建议的研究的第一部分旨在表明,许多类遍历措施所产生的几何结构上的理想边界的泊松边界。我们也感兴趣的群体不是非积极的弯曲,但有一些共同的特点,如映射类群体或一个圆的同构群。这是PI与R一起工作的自然结果。穆奇尼克第二个建议的研究方向检查重心方法作为一种工具,了解流形承认非平凡映射到非正弯曲的流形。这是微分几何在边界测度研究中的应用。通过将流形的体积和大尺度几何联系起来,我们希望利用这些方法实现Mostow刚性的进一步推广。数学和物理科学的许多显著发展揭示了给定系统中的随机过程如何经常反映该系统的某些结构特征。例如,一只蚂蚁在欧几里德平面上向北、向南、向东或向西随机移动一个单位,最终将以概率的确定性返回到它的起点。然而,当一个人允许一个额外的运动自由度时,比如说在第三维中上下运动,这就不再成立了。因此,这种“随机游走”的递归性质检测周围空间的维度。我们建议研究的灵活性之间的几何基础空间和某些随机过程的连接。我们特别感兴趣的是了解当广义随机游动的重要家庭的空间可以产生一个规定的一组测量。从另一个角度来看,这些辅助测量本身捕捉了这些空间的其他内在方面,有时可以表明空间的“刚性”。这是指空间之间的弱等价意味着强等价的现象。我们希望能发现新的僵化方式。
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)
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Christopher Connell其他文献
Minimal entropy rigidity for foliations of compact spaces
- DOI:
10.1007/bf02785426 - 发表时间:
2002-12-01 - 期刊:
- 影响因子:0.800
- 作者:
Jeffrey Boland;Christopher Connell - 通讯作者:
Christopher Connell
A Characterization of Homogeneous Spaces with Positive Hyperbolic Rank
- DOI:
10.1023/a:1020307604978 - 发表时间:
2002-01-01 - 期刊:
- 影响因子:0.500
- 作者:
Christopher Connell - 通讯作者:
Christopher Connell
Christopher Connell的其他文献
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{{ truncateString('Christopher Connell', 18)}}的其他基金
REU Site: Research Experiences for Undergraduates in Mathematics at Indiana University
REU 网站:印第安纳大学数学本科生的研究经验
- 批准号:
1757857 - 财政年份:2018
- 资助金额:
$ 12.36万 - 项目类别:
Standard Grant
REU Site: Research Experiences for Undergraduates in Mathematics at Indiana University
REU 网站:印第安纳大学数学本科生的研究经验
- 批准号:
1461061 - 财政年份:2015
- 资助金额:
$ 12.36万 - 项目类别:
Continuing Grant
Bloomington Geometry Workshop, April 26-27, 2014
布卢明顿几何研讨会,2014 年 4 月 26-27 日
- 批准号:
1430485 - 财政年份:2014
- 资助金额:
$ 12.36万 - 项目类别:
Standard Grant
Geometric rigidity for maps, foliations, and boundary structures of nonpositively curved spaces
非正弯曲空间的地图、叶状结构和边界结构的几何刚性
- 批准号:
0420432 - 财政年份:2003
- 资助金额:
$ 12.36万 - 项目类别:
Standard Grant
Geometric rigidity for maps, foliations, and boundary structures of nonpositively curved spaces
非正弯曲空间的地图、叶状结构和边界结构的几何刚性
- 批准号:
0306594 - 财政年份:2003
- 资助金额:
$ 12.36万 - 项目类别:
Standard Grant
Mathematical Sciences Postdoctoral Research Fellowship
数学科学博士后研究奖学金
- 批准号:
9902395 - 财政年份:1999
- 资助金额:
$ 12.36万 - 项目类别:
Fellowship Award
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