Harmonic Maps, Geometric Rigidity, and Non-Abelian Hodge Theory

在日常使用中，地图是地球特征的表示，提供了一种直接有效的方式来传达有关地点之间的大小、形状和距离的信息，说明了地球上元素的空间排列。地球是一个可以测量角度和距离的几何空间的例子。另一个例子是欧几里得空间用来模拟我们的物理世界。此外，非欧几里得或黎曼空间有许多应用，特别是在宇宙学中，允许科学家描述宇宙的大尺度结构，曲率和拓扑结构，并在机器人技术中，为工程师提供了一个数学框架来模拟机器人系统的运动范围。就像制图师绘制地图是为了揭示地球上各个地区的空间信息一样，数学家绘制几何空间之间的地图是为了发现有趣的特征并分析它们的几何形状。在这个研究项目中，首席研究员（PI）将专注于谐波图，这是一种特殊类型的图，可以将几何空间之间的特定能量最小化。通过分析谐波图，PI旨在揭示各种几何空间的基本属性，有助于更好地理解自然世界。谐波图的数学理论在医学（例如，医学成像）和计算机科学（例如，计算机视觉）中具有实际应用，具有进一步科学进步和社会福利的潜力。此外，该项目还包括教育方面的内容，为研究生、博士后和早期职业数学家提供指导和支持，特别是那些在STEM领域代表性不足的数学家。调和映射理论是数学家和物理学家非常感兴趣的理论，它在几何分析中起着至关重要的作用，它作为分析对象，结合了给定空间的几何、拓扑和代数信息。涉及谐波映射的技术已经成功地应用于各种数学环境中，在刚性问题、Hodge理论和teichm<s:1> ller理论中产生了重要的结果。首席研究员（PI）将继续发展谐波映射理论，其动机是其在不同数学领域的潜在应用。本文主要围绕三个关键领域展开：(1)非阿贝尔Hodge理论，旨在建立光滑拟投影变体上的非阿贝尔Hodge理论，并将Kahler流形的拓扑数据与其全纯结构联系起来；(2)几何刚度,探讨几何方法研究了刚性半单李群的晶格,尤其是不均匀晶格的晶格与非紧流形与相关研究表示在等距组光滑和奇异空间;(3)调和映射的正则性，主要研究调和映射成奇异目标的正则性理论，以更好地理解奇异集及其潜在的应用。该研究旨在推进谐波映射理论，揭示在各个数学领域具有实际意义的新见解。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。

In everyday use, maps are representations of Earth's characteristics, providing a straightforward and efficient way to convey information about sizes, shapes, and distances between places, illustrating the spatial arrangement of elements on Earth. Earth is an example of a geometric space where angles and distances can be measured. Another example is Euclidean spaces used to model our physical world. Additionally, non-Euclidean or Riemannian spaces have numerous applications, notably in cosmology, allowing scientists to describe the large-scale structure, curvature, and topology of the universe, and in robotics, providing engineers with a mathematical framework to model the range of motion of robotic systems. Just as cartographers construct maps to reveal spatial information about regions on Earth, mathematicians construct maps between geometric spaces to uncover interesting features and analyze their geometry. In this research project, the principal investigator (PI) will focus on harmonic maps, a special type of maps that minimize a specific measure of energy between geometric spaces. By analyzing harmonic maps, the PI aims to uncover essential properties of various geometric spaces, contributing to a better understanding of the natural world. The mathematical theory of harmonic maps has practical applications in medicine (e.g., medical imaging) and computer science (e.g., computer vision), with potential to further scientific progress and societal welfare. Moreover, the project includes an educational aspect, offering guidance and support to graduate students, post-docs, and early-career mathematicians, especially those underrepresented in STEM fields.Harmonic map theory holds significant interest for mathematicians and physicists, playing a vital role in geometric analysis by serving as analytical objects that incorporate geometric, topological, and algebraic information of a given space. The techniques involving harmonic maps have been successfully applied in various mathematical contexts, yielding important results in rigidity problems, Hodge theory, and Teichmüller theory. The principal investigator (PI) will continue developing the theory of harmonic maps, motivated by its potential applications in different mathematical fields. The primary focus of the proposal revolves around three key areas: (1) Non-Abelian Hodge Theory, aiming to develop non-abelian Hodge theory over smooth quasi-project varieties and connecting topological data of Kahler manifolds to their holomorphic structure; (2) Geometric Rigidity, exploring a geometric approach to study the rigidity of lattices in semisimple Lie groups, particularly non-uniform lattices associated with non-compact manifolds and studying representations of lattices in isometry groups of smooth and singular spaces; (3) Regularity of Harmonic Maps, concentrating on problems related to the regularity theory of harmonic maps into singular targets, with the goal of better understanding singular sets and their potential applications. The research seeks to advance the theory of harmonic maps, unveiling new insights with practical implications in various mathematical fields.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.