Geometry and dynamics in moduli spaces of surfaces

模空间弥漫在数学中。给定一个数学对象，相应的模空间将该对象可以具有的形状参数化。例如，在三角形的模空间中沿着一条路径走，就相当于观看一个三角形变形成另一个三角形的电影。通过这种方式，模空间帮助我们理解数学对象的表现形式如何相互变形。PI将研究具有一些附加几何结构的曲面的模空间，并利用这一点在解决一系列关于其几何和拓扑的长期猜想方面取得进展。所讨论的模空间允许一种作用，即一种“混淆”空间的方式，这与理解支配空间的“物理”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将把他的研究与努力吸引来自不同背景和数学天赋的学生结合起来。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低的学生，邀请研究生担任研究导师。学生也将被招募来帮助制作计算和教育材料，这些材料将被提供给公众。该项目将在PI正在进行的关于Hodge丛上的GL(2，R)作用的研究以及在相关模空间的研究中的应用方面取得进展。这些问题涉及到动力学、低维拓扑和代数几何中的各种问题，并使用了PI和合作者开发的新技术。最近的开创性工作表明，Hodge丛中某一点的每个GL(2，R)轨道闭合在周期坐标中是局部线性的，但到目前为止还不存在这些轨道闭合的分类。PI将在将超椭圆轨迹中的所有GL(2，R)轨道闭合分类方面取得进展，并将使用Hurwitz空间理论来建立产生新轨道闭合的纯组合机制。此外，利用最近关于双轨道闭包的工作，PI将揭示黎曼曲面的模空间中的全测地子流形的性质。这一探索将导致对TeichMuller空间中的复测地线何时是全纯收缩的更深入的理解。PI还将研究复杂仿射结构的模空间，以在一个猜想上取得进展，该猜想认为Hodge丛的地层在orbillold意义上是非球面的；并将编制一个程序来确定发散的Teichmuller测地线集合的Hausdorff维度。综上所述，这些项目将解决关于模空间几何的公开问题，同时为有理台球的研究带来新的曙光。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 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.