Geometry and dynamics in moduli spaces of surfaces

表面模空间中的几何和动力学

• 批准号: 2304840
• 负责人: Paul Apisa
• 金额: $ 25万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304840&HistoricalAwards=false
• 关键词: Geometry; dynamics; moduli; spaces; surfaces

Moduli spaces pervade mathematics. Given a mathematical object the corresponding moduli space parameterize the shapes that the object can have. For example, following a path in the moduli space of triangles corresponds to watching a movie of one triangle deforming into another. In this way moduli spaces helps us understanding how manifestations of a mathematical object can be deformed one to the other. The PI will investigate moduli spaces of surfaces with some additional geometric structure and use this to make advances in solving a suite of long-standing conjectures about their geometry and topology. The moduli spaces in question admit an action, i.e. a way of “mixing-up” the space, that is intimately connected to understanding the ``physics” governing the space. The PI and his collaborators have recently developed techniques for studying this action, which the PI will use to solve a series of problems. The PI will integrate his research with efforts to engage students from a diverse pool of backgrounds and mathematical talent. This will include devising computational research projects for undergraduate students, including those with minimal mathematical background, inviting graduate students to act as research mentors. Students will also be recruited to help produce computational and educational materials, which will be made available to the public.The project will make advances in PI's ongoing investigation of the GL(2, R) action on the Hodge bundle and applications to the study of associated moduli spaces. These questions connect to various problems in dynamics, low-dimensional topology, and algebraic geometry and use new techniques developed by the PI and collaborators. Recent groundbreaking work has shown that each GL(2, R) orbit closure of a point in a stratum of the Hodge bundle is locally linear in period coordinates, but as yet no classification of these orbit closures exists. The PI will make progress in classifying all GL(2, R) orbit closures in hyperelliptic loci of strata that are “sufficiently big”, and will use the theory of Hurwitz spaces to build a purely combinatorial mechanism for producing new orbit closures. In addition, using recent work on geminal orbit closures, the PI will uncover properties of totally geodesic submanifolds in the moduli space of Riemann surfaces. This inquiry will lead to a deeper understanding of when complex geodesics in Teichmuller space are holomorphic retracts. The PI will also study the moduli spaces of complex affine structures to make progress on a conjecture that strata of the Hodge bundle are aspherical in the orbifold sense; and will work on a program to determine the Hausdorff dimension of the set of divergent Teichmuller geodesic rays. Put together, these projects will resolve open questions about the geometry of moduli space, while shedding fresh light on the study of rational billiards.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.

模量空间遍布数学。给定数学对象，相应的模量空间参数化对象可能具有的形状。例如，在三角形的模量空间中遵循一条路径，对应于观看一个三角形变形的电影。通过这种方式，模量空间有助于我们了解数学对象的表现如何将一个变形为另一个。 PI将使用一些其他几何结构来研究表面的模量空间，并以此来进步解决有关其几何形状和拓扑的一系列长期猜想。所讨论的模量空间接受一个动作，即一种“混合”空间的方式，这与理解````物理''''管理空间密切相关。 PI和他的合作者最近开发了研究此操作的技术，PI将使用该技术来解决一系列问题。 PI将努力将他的研究与来自不同背景和数学才能不同的学生吸引。这将包括为本科生设计计算研究项目，包括那些数学背景最少的人，邀请研究生担任研究导师。还将招募学生以帮助制作计算和教育材料，并将其提供给公众。该项目将在PI对HODGE束的GL（2，R）行动以及对相关模量空间的研究中的应用中进行进步。这些问题与动态，低维拓扑和代数几何形状的各种问题有关，并使用PI和合作者开发的新技术。最近的开创性工作表明，霍奇束层中一个点的每个GL（2，R）轨道闭合在时期坐标中是局部线性的，但尚无对这些轨道封闭的分类。 PI将在“足够大”的地层中的所有GL（2，R）轨道闭合分类中取得进展，并将使用Hurwitz空间理论来构建纯粹的组合机制来产生新的轨道封闭。此外，使用最近在Geminal Orbit封闭的工作，PI将在Riemann表面的模量空间中发现完全大地​​测量的亚体的特性。此询问将导致对teichmuller空间中复杂的大地测量学何时是全体形态缩回的更深入的理解。 PI还将研究复杂仿射结构的模量空间，以在一个猜想中取得进展，即霍奇束的地层在Orbifold的意义上是非球体的。并将在一个程序上工作，以确定一组发散的Teichmuller Geodetic射线的Hausdorff尺寸。组合在一起，这些项目将解决有关模量空间几何形状的开放问题，同时为研究理性台球的研究提供了新的启示。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估被认为是宝贵的支持。