Curves, Counting, and Correlations

拓扑学是数学的一个领域，研究在拉伸或弯曲下保持不变的空间的那些性质。具有固定拓扑形状的曲面可以具有许多不同的几何体。例如，没有孔的二维形状是拓扑球体，它可以是圆形球体（像行星的表面），多面体（具有小平面的晶体状形状）或椭圆体（蛋形形状）。带有一个孔的形状（圆环）可能看起来像内胎、轮胎或呼拉圈。所有可能几何的集合被称为模空间，这些空间在基本物理问题中反复出现。这个NSF奖项为一个项目提供资金，该项目从理解这些表面上的曲线族的角度研究典型或随机表面的外观问题。研究和推广活动是相辅相成的;该项目突出了数学不同领域之间的桥梁，以及数学研究，指导和公众参与计划之间的深层联系。此外，该项目还为研究生提供研究培训机会。从简单的状态问题开始，计算几何形状来自欧几里得多边形和多面体的表面上的特殊轨迹，并受到理论物理的启发（牛顿力学，电子输运和超对称量子场论），该项目连接到数学的许多不同领域的基本问题，包括Teichmuller和齐次动力学;丢番图近似和几何数;概率模型和极限定理;以及曲面上亚纯微分的模空间几何。在他之前的合作工作的基础上，PI计划探索高属曲面和高维环面上不同几何形状曲线的计数和分布问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Topology is a field of mathematics that studies those properties of a space that remain invariant under stretching or bending. A surface of a fixed topological shape can have many different geometries. For example, a two-dimensional shape with no holes is a topological sphere, which could be a round sphere (like the surface of a planet), a polyhedron (a crystal like-shape with facets), or an ellipsoid (an egg-type shape). A shape with one hole (a torus) could look like an inner tube, a tire, or a hula hoop. The collection of all possible geometries is known as a moduli space, and these spaces occur over and over again in fundamental physical questions. This NSF award provides funding for a project to study questions of what typical, or random, surfaces look like, from the point of view of understanding families of curves on these surfaces. The research and outreach activities are mutually reinforcing; the project highlights bridges between different areas of mathematics, and deep connections between research, mentoring, and public engagement programs in mathematics. In addition the project provide research training opportunities for graduate students. Starting with simple to state problems about counting special trajectories on surfaces whose geometry comes from Euclidean polygons and polyhedra, and motivated by theoretical physics (Newtonian mechanics, electron transport, and supersymmetric quantum field theory), this project connects to fundamental questions in many different areas of mathematics, including Teichmuller and homogeneous dynamics; Diophantine approximation and the geometry of numbers; probabilistic models and limit theorems; and the geometry of moduli spaces of meromorphic differentials on surfaces. Building on his prior collaborative work, the PI plans to explore questions on the counting and distributions of curves in varying geometries on higher-genus surfaces and higher-dimensional tori.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.