Quantitative Homotopy Theory and Geometric Inequalities

定量同伦理论和几何不等式

• 批准号: 217655-2013
• 负责人: Rotman, Regina
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570197
• 关键词: Quantitative; Homotopy; Theory; Geometric; Inequalities

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. Among my previous results are estimates for the length of a shortest geodesic loop in terms of the diameter of a manifold as well as upper bounds for the area of minimal surfaces in terms of the diameter / volume of a manifold. I plan to work on the questions of such nature in the future.

我的兴趣领域是黎曼几何，这是一个推广的几何曲面。我特别感兴趣的是黎曼几何中的静态对象，如测地线段，测地线回路，周期测地线，静态测地线网和周期，极小曲面等。 我的大量工作旨在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，它们只是直线。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，可以一直走，只是回到起点。 如果发生这种情况，这意味着我们遇到了测地线环。 在闭黎曼流形的任何一点上，都存在无穷多个方向，这样，如果我们继续沿着这个方向走，我们将回到起点。 我们要走很远的路吗？ 它是如何连接到歧管的大小？ 这是我感兴趣的问题的一个例子。现在想象你正沿着一条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。这是如果你沿着沿着行走会发生的事情。 这意味着你遇到了一条周期测地线。 周期测地线不仅返回到起点，而且是平滑地返回。 周期测地线在几何学、分析学、动力学系统中有着特殊的作用。 极小测地线网是周期测地线的同调类似物。 类似的概念首先出现在最佳电话网络的研究中，后来又出现在其他应用中。极小曲面在物理学中被用作肥皂泡的数学模型，在弦论中也是如此。 在我以前的结果是估计的长度最短的测地线循环的直径方面的歧管以及上限的面积极小曲面的直径/体积方面的歧管。 我计划今后研究这类性质的问题。