Geometric and dynamical problems on surfaces

首席研究员将致力于解决几何学、拓扑学和曲面动力学方面的问题。 第一部分的建议的目的是增加我们的理解的映射类组，其行动上的Teichmuller空间的表面和边界在无穷远处的Teichmuller空间。在具体的问题是一个试图表明，从计数轨道点的角度来看，为行动的映射类组的Teichmuller空间，一个随机映射类组元素是伪Anosov。另一个问题是研究映射类群在Thurston测量叶理空间上的轨道分布。 该提案第二部分的目的是研究平移表面上的流动动力学。 主要研究者特别感兴趣的现象，最小的，但不是唯一的遍历方向的线性流动的翻译表面。主要研究者的工作在几个不同领域的理论数学。这些领域是拓扑学、几何学和动力系统。他们在研究表面性质的广泛领域走到了一起。这是一个可以追溯到几百年前的数学主题，但仍然是现代研究的一个充满活力的领域。主要研究员的工作涉及扩大我们对这些领域的知识，也涉及培养研究生成为研究数学家。

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.