Quantitative Homotopy Theory and Geometric Inequalities

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，如测地线段、测地线环、周期测地线、静止测地线网和圈、极小曲面等。我的大量工作旨在分析各种极端物体的“大小”与环境空间的“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，笔直行走是可能的，只是回到起点。如果发生这种情况，就意味着我们遇到了测地线回路。在闭黎曼流形的任何一点都存在无限多个方向，这样，如果我们继续沿着这个方向走，我们就会回到起点。我们要走很长一段路吗？它是如何与歧管的大小联系起来的？这是我感兴趣的问题的一个例子。现在想象你沿着流形上的一条测地线行走，这意味着你的方向没有改变，一段时间后，你注意到你正沿着同一条路径走。如果你沿着赤道走，就会发生这种情况。这意味着你遇到了一条周期测地线。周期测地线不仅会返回到起点，而且会平滑地返回起点。周期测地线在几何、分析、动力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比。类似的概念首先出现在对最优电话网络的研究中，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，在弦理论中也是如此。在我之前的结果中，有用流形的直径来估计最短测地线圈的长度，以及用流形的直径/体积来估计最小曲面面积的上界。我计划在未来致力于这类性质的问题。

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.