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

映射类群的曲面和曲面子群的模空间中的轨道闭包

• 批准号: 1856155
• 负责人: Alexander Wright
• 金额: $ 32万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856155&HistoricalAwards=false
• 关键词: Orbit; Closures; Moduli; Spaces; Surfaces

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.

曲面的模空间在整个数学和理论物理中起着中心作用。每个这样的模空间都可以被认为是一个曲面所有可能形状的宇宙。正如万有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）作用的定律规定了这些宇宙中所有可能表面的表面轨道。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，并将提供关于表面本身的深入信息。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形状。该项目的第二部分旨在深入了解映射类组，它编码了将表面包裹到自身上的所有方法，以及称为表面束的四维形状。本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训练基地，并与首席研究员正在进行的进一步推广Maryam Mirzakhani在这一领域鼓舞人心的工作的努力相联系。具体来说，PI建议在两个相关的项目中从事Teichmueller理论和动力学交叉的研究。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。首席研究员建议对具有大秩的轨道闭包进行分类，采用由首席研究员和他的合作者最近提出的关于圆柱体变形和退化的定理，以及Eskin-Mirzakhani-Mohammadi的开创性结果。在第二个程序中，主要研究者提出利用关于Teichmueller测地线流的动力学和几何结果来研究映射类群的凸紧曲面子群。这种曲面子群的存在性等价于具有Gromov双曲基群的曲面上的曲面束的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Strongly obtuse rational lattice triangles

强钝角有理格子三角形

• DOI: 10.1090/tran/8415
• 发表时间: 2021
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Larsen, Anne;Norton, Chaya;Zykoski, Bradley
• 通讯作者: Zykoski, Bradley

Periodic points on the regular and double n-gon surfaces

规则和双 n 边​​形表面上的周期点

• DOI: 10.1007/s10711-022-00730-6
• 发表时间: 2022
• 期刊: Geometriae Dedicata
• 影响因子: 0.5
• 作者: Apisa, Paul;Saavedra, Rafael M.;Zhang, Christopher
• 通讯作者: Zhang, Christopher

Hodge and Teichmüller

霍奇和泰希米勒

• DOI: 10.3934/jmd.2022007
• 发表时间: 2022
• 期刊: Journal of Modern Dynamics
• 影响因子: 1.1
• 作者: Kahn, Jeremy;Wright, Alex
• 通讯作者: Wright, Alex