Geometry and Dynamics of Moduli Spaces of Surfaces

这一建议的目的是进一步探索和理解曲面上的不同几何结构和相应模空间上的自然诱导动力系统。我们打算研究给定亏格的黎曼曲面的模空间的余切丛上的自然几何流的动力学。这一领域的大多数结果和问题都是基于齐次空间上这些流与幂等流之间的类比。我们还计划研究当g趋于无穷大时，亏格g的Riemann曲面的模空间上重言式线丛的交线对的渐近行为。这可能有助于揭示大亏格随机双曲曲面在不同模型中的几何性质。我们提出的主要问题集中在Riemann曲面的几何及其变形上。黎曼曲面的模空间在不同的背景下自然出现，例如在代数几何和弦理论中。这一研究方案是跨学科的，涉及不同的数学分支，特别是动力系统、低维拓扑学、代数几何、双曲几何和几何群论。

The goal of this proposal is to further explore and understand different geometric structures on surfaces and natural induced dynamical systems on the corresponding moduli spaces. We intend to study the dynamics of natural geometric flows on the cotangent bundle of the moduli space of Riemann surfaces of a given genus. Most of the results and questions in this area are based on analogies between these flows and unipotent flows on homogeneous spaces. We also plan to investigate the asymptotic behavior of intersection pairings of tautological line bundles on the moduli spaces of Riemann surfaces of genus g as g goes to infinity. This could shed some light on geometric properties of random hyperbolic surfaces of large genus in different models.The main problems motivating our proposed project are centered around the geometry of Riemann surfaces and their deformations. Moduli spaces of Riemann surfaces appear naturally in various contexts, for example in algebraic geometry and string theory. This research proposal is interdisciplinary and has connections to different branches of mathematics, especially dynamical systems, low dimensional topology, algebraic geometry, hyperbolic geometry and geometric group theory.