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。每一个这样的模空间都可以被认为是一个表面可以采取的所有可能形状的宇宙。正如引力定律规定了我们宇宙中行星和恒星的轨道一样，另一套称为GL+（2，R）作用量的定律规定了这些宇宙中所有可能表面的轨道。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一个可能的行为表，并将给出关于表面本身的深层信息。这些信息可以用于涉及动力学和多边形形状的更具体的数学和物理应用。这个项目的第二部分旨在深入了解映射类组，它编码了所有将曲面包裹到自身上的方式，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，研究生和博士后学生提供了理想的培训基地，并与主要研究者正在进行的努力相联系，以进一步推广Maryam Mirzakhani在这一领域的鼓舞人心的工作。具体来说，PI建议在Teichmueller理论和动力学的交叉点进行研究，在两个相关的程序。第一个程序涉及GL+（2，R）二次和阿贝尔微分的轨道闭包，也称为（半）平移曲面。主要研究者建议分类轨道闭合有大的排名，通过采用最近的定理关于圆柱体变形和退化的主要研究者和他的合著者，以及开创性的结果Eskin Mirzakhani-Mohammadi。在第二个程序中，主要研究者提出调查凸的映射类组的协紧表面子群，采用动力学和几何结果有关Teichmueller测地线流。这种表面子群的存在被认为是等价于存在的表面丛表面与Gromov双曲基本group.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.