Topics in Low Dimensional Geometric Analysis

拓扑学的许多进步都源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。对3流形的研究本质上是对宇宙可能形状的研究，因此自然处于数学和理论物理的交叉点。表面是二维空间，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，这一研究处于数学各个领域的交叉点。给定曲面上所有几何结构的参数空间称为模空间。我们学习模空间是因为要理解空间中的一个特定结构我们必须理解所有结构的空间结构。调和映射理论起源于最小曲面理论。它旨在建立谐波映射的重要性质，从而提供关于最小表面的有用信息，例如跨越线边界的肥皂膜的物理描述。首席研究员将研究Teichmueller空间的几何、曲面群表示的等变调和映射的存在性、曲面到3流形的映射以及不可压缩流体的二维欧拉方程的解。主要目标是：在Caratheodory和Teichmueller指标一致的地方对Teichmueller盘进行分类；在证明舍恩猜想的基础上，建立了具有边界的紧曲面和无穷秩曲面的基本群的表示所对应的等变调和映射的存在性的结果；研究双曲型3流形的简单环猜想考虑二维欧拉方程中涡度梯度能否实现双指数增长。该奖项反映了美国国家科学基金会的法定使命，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，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.