Geometry and dynamics in moduli spaces of surfaces

模空间在数学中无处不在。给定一个数学对象，相应的模空间参数化了该对象可能具有的形状。 例如，在三角形的模空间中沿着一条路径走，相当于观看一部三角形变形为另一个三角形的电影。模空间以这种方式帮助我们理解数学对象的表现形式是如何从一个变形到另一个的。 PI将研究具有一些额外几何结构的曲面的模空间，并使用它来解决一系列关于其几何和拓扑的长期问题。所讨论的模空间承认一种作用，即一种“混合”空间的方式，它与理解支配空间的“物理学”密切相关。PI和他的合作者最近开发了研究这种行为的技术，PI将使用这些技术来解决一系列问题。 PI将整合他的研究，努力吸引来自不同背景和数学人才的学生。 这将包括为本科生设计计算研究项目，包括那些具有最低数学背景的学生，邀请研究生担任研究导师。 该项目还将招募学生帮助制作计算和教育材料，这些材料将提供给公众。该项目将推动PI正在进行的关于Hodge丛上GL（2，R）作用的研究，并将其应用于相关模空间的研究。这些问题与动力学、低维拓扑和代数几何中的各种问题相联系，并使用PI和合作者开发的新技术。最近的突破性工作表明，每个GL（2，R）轨道封闭的一个点在一个阶层的霍奇丛是局部线性的周期坐标，但尚未分类这些轨道封闭存在。PI将在分类所有GL（2，R）轨道闭合在超椭圆轨迹的地层是“足够大”的进展，并将使用赫维茨空间的理论，以建立一个纯粹的组合机制，产生新的轨道闭合。此外，利用最近的工作对Geminal轨道闭包，PI将揭示性质的全测地子流形的模空间的黎曼曲面。这一调查将导致更深入的了解时，复测地线在Teichmuller空间是全纯收缩。 PI还将研究复杂仿射结构的模空间，以便在霍奇丛的地层在轨道意义上是非球面的猜想上取得进展;并将制定一个程序来确定发散的Teichmuller测地线的Hausdorff维数。总而言之，这些项目将解决关于模数空间几何的公开问题，同时为理性台球的研究提供新的思路。该奖项反映了NSF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。

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.