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大小，形状和地点之间距离的信息，说明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。另一个例子是用于模拟我们的物理世界的欧几里得空间。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以揭示地球上区域的空间信息一样，数学家在几何空间之间绘制地图以揭示有趣的特征并分析其几何形状。在这个研究项目中，主要研究者（PI）将专注于调和映射，这是一种特殊类型的映射，可以最小化几何空间之间的能量的特定度量。通过分析调和映射，PI旨在揭示各种几何空间的基本性质，有助于更好地理解自然世界。调和映射的数学理论在医学中有实际应用（例如，医学成像）和计算机科学（例如，计算机视觉），具有促进科学进步和社会福利的潜力。此外，该项目还包括一个教育方面，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。调和映射理论对数学家和物理学家有着重要的兴趣，在几何分析中发挥着至关重要的作用，它作为分析对象，结合了给定空间的几何、拓扑和代数信息。涉及调和映射的技术已成功地应用于各种数学背景，在刚性问题、霍奇理论和泰希米勒理论中产生了重要结果。主要研究者（PI）将继续发展调和映射理论，其动机是它在不同数学领域的潜在应用。该提案的主要重点围绕三个关键领域：（1）非阿贝尔Hodge理论，旨在发展光滑拟投影簇上的非阿贝尔Hodge理论，并将Kahler流形的拓扑数据与其全纯结构联系起来;（2）几何刚性，探索半单李群中格刚性的几何研究方法，特别是与非紧流形相关的非均匀格，并研究光滑和奇异空间的等距群中的格的表示;（3）调和映射的正则性，主要研究奇异目标下调和映射的正则性理论，其目的是更好地理解奇异集及其潜在应用。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，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.