Topics in Low Dimensional Geometric Analysis

拓扑学的许多进展都来自于几何学和动力学思想的应用。本项目旨在继续这一调查路线。更具体地说，我们的目标是研究3-流形，调和映射和黎曼曲面。三维流形的研究本质上是对宇宙可能形状的研究，因此自然地处于数学和理论物理的交叉点。 曲面是二维空间，就像球或甜甜圈的表面。它们的拓扑和几何性质已被广泛研究，这项研究是在所有数学领域的交叉点。给定曲面上所有几何结构的参数空间称为模空间。我们研究模空间是因为要理解空间上的一个特定结构，我们必须理解所有结构的空间的结构。 调和映射理论起源于极小曲面理论。该课程旨在建立调和映射的重要性质，从而提供有关极小曲面的有用信息，如跨越导线边界的肥皂膜的物理描述。主要研究者将研究Teichmueller空间的几何，曲面群表示的等变调和映射的存在性，从曲面到3-流形的映射，以及不可压缩流体的2D-Euler方程的解。主要目标是：在Caratheodory度量和Teichmueller度量一致的情况下，对Teichmueller圆进行分类;在Schoen猜想证明的基础上，建立了与有边界的紧致曲面和无限秩曲面的基本群表示对应的等变调和映射的存在性结果;研究了双曲3-流形的单圈猜想;考虑二维欧拉方程中的涡度梯度是否可以实现双指数增长。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而被认为值得支持。

Numerous advances in topology have arisen from applications of ideas from geometry and dynamics. This project aims to continue this line of investigation. More specifically, we aim to study 3-manifolds, harmonic maps, and Riemann surfaces. The study of 3-manifolds is in essence the study of the possible shapes of the universe and as such sits naturally at the intersection of mathematics and theoretical physics. Surfaces are two-dimensional spaces like the surface of a ball or a doughnut. Their topological and geometric properties have been extensively studied and this research lies at the intersection of all fields of mathematics. The parameter space of all geometric structures on a given surface is called the moduli space. We study moduli spaces because to understand one particular structure on a space we must understand the structure of the space of all structures. Theory of harmonic maps originates in the theory of minimal surfaces. It aims to establish important properties of harmonic maps which in turn provide useful information about minimal surfaces like the physical description of a soap film spanning a wire boundary.The principal investigator will study geometry of Teichmueller spaces, existence of equivariant harmonic maps for surface group representations, maps from surfaces into 3-manifolds, and the solutions of the 2D-Euler equation for incompressible fluids. The main goals are to: Classify Teichmueller discs where the Caratheodory and Teichmueller metrics agree; To build on the proof of the Schoen conjecture and establish results about the existence of equivariant harmonic maps corresponding to representations of the fundamental groups of compact surfaces with boundary and of surfaces of infinite rank; To investigate the simple loop conjecture for hyperbolic 3-manifolds; To consider whether the gradient of the vorticity in the 2D-Euler equation can achieve the double exponential growth.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.