Minimal surfaces and quantitative topology of the space of cycles

建议的研究计划涉及的存在性和性质的最小和常曲率曲面的黎曼流形。肥皂泡和带电粒子在磁场中的轨迹是这些现象的真实的例子。更一般地说，极小曲面理论中的问题与偏微分方程、广义相对论、工程和许多其他领域中的许多问题密切相关。 在1960年代，Almgren提出了一种新的技术来构造闭流形中的极小曲面，现在被称为Almgren-Pitts Min-Max Theory。在过去的几年里，最小-最大理论经历了一次复兴。马奎斯和内维斯对威玛猜想的证明是一个了不起的成就，随后又有许多其他重要的结果，回答了一些长期存在的问题。目前的研究计划是集中在一些基本问题的家庭的周期连接的问题，在最小最大理论和应用的几何和拓扑。这个研究项目是在接口的几何变分法和定量拓扑学。它包含了一些问题有关的几何性质的空间平坦的周期在黎曼流形和存在的家庭的周期满足某些特殊条件的目标是获得信息的正则性，几何和拓扑的最小最大极小超曲面。该项目依赖于Gromov，Guth，Marques和Neves开发的技术和思想，以及拓扑学，几何测量理论和分析的其他思想。

The proposed research program concerns existence and properties of minimal and constant curvature surfaces in Riemannian manifolds. Soap bubbles and trajectories of charged particles in a magnetic field are some of the real world examples of these phenomena. More generally, problems in minimal surface theory are closely related to many problems in partial differential equations, general relativity, engineering and many other areas.  In 1960s Almgren suggested a new technique for the construction of minimal surfaces in closed manifolds, which is now known as the Almgren-Pitts Min-Max Theory. In the last several years the Min-Max Theory has experienced a renaissance. The proof of the Willmore conjecture by Marques and Neves was one remarkable achievement with many other important results that followed, answering a number of long-standing conjectures. Current research proposal is focused on some fundamental questions about families of cycles in connection with problems in Min-Max Theory and with applications in geometry and topology. This research project lies at the interface of Geometric calculus of variations and quantitative topology. It contains a number of problems related to the geometric properties of the space of flat cycles in a Riemannian manifold and existence of families of cycles satisfying certain special conditions with the goal of obtaining information about regularity, geometry and topology of min-max minimal hypersurfaces. The project relies on techniques and ideas developed by Gromov, Guth, Marques and Neves, as well as other ideas from topology, geometric measure theory and analysis.