Geometry and Dynamics of Moduli Spaces of Surfaces

这个提议的目标是进一步探索和理解不同的几何结构的表面和自然诱导动力系统相应的模空间。本文研究了亏格黎曼曲面模空间余切丛上自然几何流的动力学性质。这一领域的大多数结果和问题都是基于这些流与齐次空间上的幂幺流之间的类比。我们还计划研究重言式线丛在亏格为g的黎曼曲面的模空间上的交对在g趋于无穷大时的渐近行为。这将有助于揭示大亏格随机双曲曲面在不同模型下的几何性质。激发我们提出这个项目的主要问题是围绕Riemann曲面的几何及其变形。黎曼曲面的模空间自然地出现在各种背景下，例如代数几何和弦理论。该研究计划是跨学科的，与数学的不同分支有联系，特别是动力系统，低维拓扑，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.