Orbit Closures in Moduli Spaces of Surfaces and Surface Subgroups of Mapping Class Groups

曲面的模空间在整个数学和理论物理中起着核心作用。每个这样的模空间都可以被认为是一个宇宙，具有一个曲面可以采用的所有可能的形状。就像引力定律规定了我们宇宙中行星和恒星的轨道一样，另一组不同的定律，称为GL(2，R)作用量，规定了这些宇宙中所有可能的表面的表面的轨道。该项目的第一部分旨在获得这些轨道的分类结果。这样的分类结果将提供可能的行为的表，并将提供关于曲面本身的深入信息。这些信息可以用于更具体的数学和物理应用，包括动力学和多边形。本项目的第二部分旨在深入了解映射类组，它编码了将曲面包裹到自身的所有方式，以及称为曲面束的四维形状。这项研究的各个方面为本科生、研究生和博士后提供了一个理想的培训场所，并与首席研究员进一步普及Maryam Mirzakhani在这一领域的鼓舞人心的工作的持续努力相联系。具体地说，PI建议在两个相关的项目中从事Teichmueller理论和动力学的交叉点的研究。第一个程序涉及二次和阿贝尔微分的GL(2，R)轨道闭包，也称为(半)平移曲面。主要研究人员建议利用主要研究人员和他的合著者最近发展的关于圆柱体变形和退化的定理以及Ekin-Mirzakhani-Mohammadi的开创性结果来对具有大秩次的轨道闭合进行分类。在第二个程序中，主要研究者建议利用关于Teichmueller测地线流的动力学和几何结果来研究映射类群的凸余紧曲面子群。这种表面子群的存在被认为等同于具有格罗莫夫双曲基本群的表面丛的存在。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Moduli spaces of surfaces play a central role throughout mathematics and theoretical physics. Each such moduli space can be thought of as a universe of all possible shapes that a surface can take. Just as the laws of gravity prescribe the orbits of the planets and stars of our universe, a different set of laws, called the GL+(2,R) action, prescribe orbits of surfaces in these universes of all possible surfaces. The first part of this project aims to obtain classification results for these orbits. Such classification results would provide a table of possible behaviors, and would give deep information about the surfaces themselves. This information could be used in more concrete mathematical and physical applications involving dynamics and polygonal shapes. The second part of this project aims to give insight into the mapping class group, which encodes all ways of wrapping a surface onto itself, as well as four dimensional shapes called surface bundles. Aspects of this research provide an ideal training ground for undergraduate, graduate, and postdoctoral students, and connect to the principal investigator's ongoing efforts to further popularize Maryam Mirzakhani's inspiring work in this area. Specifically, the PI proposes to engage in research at the intersection of Teichmueller theory and dynamics, in two related programs. The first program concerns GL+(2,R) orbit closures of quadratic and Abelian differentials, also known as (half) translation surfaces. The principal investigator proposes to classify orbit closures that have large rank, by employing recent theorems concerning cylinder deformations and degenerations developed by the principal investigator and his coauthors, as well as the seminal results of Eskin-Mirzakhani-Mohammadi. In the second program, the principal investigator proposes to investigate convex cocompact surface subgroups of the mapping class group, employing dynamical and geometric results concerning the Teichmueller geodesic flow. The existence of such surface subgroups is known to be equivalent to the existence of surface bundles over surface with Gromov hyperbolic fundamental group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.