Geometric and dynamical problems on surfaces

首席研究员将致力于研究曲面的几何、拓扑和动力学问题。建议的第一部分的目的是增加我们对映射类群的理解，它在曲面的TeichMuller空间和TeichMuller空间无穷远处的边界上的作用。在这些具体问题中，有一个问题试图证明，从计算TeichMuller空间上映射类群作用的轨道点的观点来看，随机映射类群元是伪Anosov的。另一个问题是研究映射类群在测得的叶状空间上的轨道分布。该提案的第二部分的目的是研究平移表面上的流动动力学。主要研究人员对平移曲面上的线性流的极小但不唯一的遍历方向现象特别感兴趣。主要研究人员在几个不同的理论数学领域工作。这些领域是拓扑学、几何学和动力学系统。他们聚集在研究表面性质的广泛领域。这是一个可以追溯到几百年前的数学学科，但仍然是现代研究的一个充满活力的领域。首席研究员的工作包括扩展我们在这些领域的知识，还包括培养研究生成为研究型数学家。

The principal investigator will work on problems in the geometry, topology, and dynamics of surfaces. The objective of the first part of the proposal is to increase our understanding of the mapping class group, its action on the Teichmuller space of a surface and the boundary at infinity of Teichmuller space. Among the specific problems is one that attempts to show that from the point of view of counting orbits points for the action of the mapping class group on Teichmuller space, a random mappping class group element is pseudo-Anosov. Another problem is to study the distribution of orbits of the mapping class group on Thurston's space of measured foliations. The objective of the second part of the proposal is to study the dynamics of flows on translation surfaces. The principal investigator is interested specifically in the phenomenon of minimal but not uniquely ergodic directions for the linear flow on the translation surface.The principal investigator works in several diverse areas of theoretical mathematics. These fields are topology, geometry, and dynamical systems. They come together in the broad area of studying properties of surfaces. This is a subject of mathematics that goes back hundreds of years, and yet remains a vibrant area of modern research. The work of the principal investigator involves expanding our knowledge of these areas and also involves the training of graduate students to become research mathematicians.