Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
基本信息
- 批准号:217655-2013
- 负责人:
- 金额:$ 1.68万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2013
- 资助国家:加拿大
- 起止时间:2013-01-01 至 2014-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
My field of interest is Riemannian Geometry, which is a generalization of the geometry of surfaces. I specifically am interested in stationary objects in Riemannian geometry, such as geodesic segments, geodesic loops, periodic geodesics, stationary geodesic nets and cycles, minimal surfaces, etc. A great deal of my work is aimed at analyzing the connection between the "size'' of various extremal objects and the "size" of the ambient space, as well as their existence. Geodesics are the "straightest" curves on a manifold. For example, in a Euclidean geometry, they are simply the straight lines. On the standard sphere, geodesics are the big circles, like the equator of the sphere. So, on a manifold, it is possible to walk straight, only to return to the starting point. If this happens, it means that we have encountered a geodesic loop. At any point of a closed Riemannian manifold there exists infinitely many directions, such that we will return to the starting point, if we continue walking in this direction. Will we have to walk a long distance? How does it connect to the size of the manifold? This is an example of questions that are of interest to me. Now imagine that you are walking along a geodesic on a manifold, which means that your direction is not changing, and after some time you notice that you are going along the same path. This is something that would happen if you walk along the equator. That means that you have encountered a periodic geodesic. A periodic geodesic does not only return to the starting point, but does so smoothly. Periodic geodesics play a special role in Geometry, Analysis, Dynamical Systems. Minimal geodesic nets are homological analogs of periodic geodesics. Similar concepts were first encountered in a study of optimal telephone networks, and later appeared in other applications. Minimal surfaces are used in Physics as mathematical models of soap bubbles, as well as in String Theory.
我感兴趣的领域是黎曼几何,它是曲面几何的推广。我对黎曼几何中的静止物体特别感兴趣,例如测地线段,测地线回路,周期测地线,静止测地线网和循环,最小表面等。我的很多工作都是为了分析各种极端物体的“大小”与周围空间的“大小”之间的联系,以及它们的存在。测地线是流形上“最直”的曲线。例如,在欧几里得几何中,它们就是简单的直线。在标准球面上,测地线是大圆,就像球面的赤道。因此,在流形上,有可能直线行走,但只能返回到起点。如果发生这种情况,就意味着我们遇到了一个测地线回路。在封闭黎曼流形的任何一点上,存在无限多个方向,如果我们继续沿着这个方向走,我们将回到起点。我们要走很长的一段路吗?它如何与歧管的尺寸相连接?这是我感兴趣的问题的一个例子。现在想象一下,你正沿着流形上的测地线行走,这意味着你的方向没有改变,过了一段时间,你注意到你正沿着同一条路径行走。如果你沿着赤道行走,这种情况就会发生。这意味着你遇到了一个周期测地线。周期测地线不仅可以返回到起点,而且可以平滑地返回。周期测地线在几何、分析、动力系统中起着特殊的作用。极小测地线网是周期测地线的同源类似物。类似的概念最初是在对最佳电话网络的研究中遇到的,后来出现在其他应用中。最小表面在物理学中被用作肥皂泡的数学模型,在弦理论中也是如此。
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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Rotman, Regina其他文献
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{{ truncateString('Rotman, Regina', 18)}}的其他基金
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
RGPIN-2018-04523 - 财政年份:2022
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
RGPIN-2018-04523 - 财政年份:2021
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
RGPIN-2018-04523 - 财政年份:2020
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
RGPIN-2018-04523 - 财政年份:2019
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
RGPIN-2018-04523 - 财政年份:2018
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
217655-2013 - 财政年份:2017
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
217655-2013 - 财政年份:2016
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
217655-2013 - 财政年份:2015
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
- 批准号:
217655-2013 - 财政年份:2014
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
External objects in Riemannian geometry
黎曼几何中的外部对象
- 批准号:
217655-2008 - 财政年份:2012
- 资助金额:
$ 1.68万 - 项目类别:
Discovery Grants Program - Individual
相似海外基金
Quantitative Homotopy Theory and Geometric Inequalities
定量同伦理论和几何不等式
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RGPIN-2018-04523 - 财政年份:2022
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$ 1.68万 - 项目类别:
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RGPIN-2018-04523 - 财政年份:2018
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$ 1.68万 - 项目类别:
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The length of the shortest closed geodesic and the quantitative homotopy theory
最短闭合测地线长度与定量同伦理论
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