Harmonic Maps into Spaces with an Upper Curvature Bound

“地图”这个词的日常定义是“陆地或海洋区域的图示，显示物理特征、城市、道路等。”制图员构建地图是为了揭示一个地理区域的有趣空间信息。以类似的方式，数学家在几何空间之间构建地图，以发现这些空间的有趣特征。这里所说的几何空间，我们指的是配备了角度、距离、面积等概念的空间。这些空间包括欧几里德空间，它经常被用来模拟我们日常的物理世界，或者非欧几里德空间或黎曼空间，它可以用作我们宇宙的大规模模型。该奖项提供资金，用于研究几何空间之间的特殊地图，称为调和地图，最大限度地减少特定的能量概念。通过分析调和映射，PI旨在揭示重要几何空间的属性，这些属性将导致对自然世界的更好理解。调和映射的数学理论已经应用于医学(例如医学成像)和计算机科学(例如计算机视觉)等各个领域，并具有进一步的潜在应用，有助于我们社会的科学进步和福利。该项目还包括一个教育部分，并通过向研究生、博士后和职业生涯初期的数学家，特别是那些在STEM领域任职人数不足的人提供教学和咨询，支持多样性。该项目主要研究具有曲率上界的空间中的调和映射。黎曼流形之间的调和映射是一类椭圆型偏微分方程组(调和映射方程)的解，也是一个涉及狄利克雷能量的变分问题的解。自从现代几何分析作为一门核心数学学科出现以来，调和映射理论一直处于该领域的前沿，并不断发现重要的应用。最近的发展是研究到满足曲率上界的完备度量空间中的调和映射。这个项目的目标是发展这一理论，以便将其应用于解决刚性问题，了解Kahler流形和射影变种上的表面纤维结构，并研究准共形映射。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The everyday definition of the word “map” is “a diagrammatic representation of an area of land or sea showing physical features, cities, roads, etc.” Cartographers construct maps to reveal interesting spacial information about a geographical region. In a similar way, mathematicians construct maps between geometric spaces to discover interesting features of those spaces. Here, by a geometric space, we mean a space equipped with notions of angles, distances, areas, etc. They include Euclidean spaces which are often used to model our everyday physical world or non-Euclidean or Riemannian spaces which can be used as a large scale model of our universe. This award provides funding to study special maps between geometric spaces called harmonic maps that minimize a certain notion of energy. By analyzing harmonic maps, the PI aims to uncover properties of important geometric spaces that would lead to a greater understanding of the natural world. The mathematical theory of harmonic maps has been applied in diverse fields such as medicine (for example, in medical imaging) and computer science (for example, in computer vision), and has further potential applications aiding in the scientific progress and welfare of our society. The project also has an educational component and supports diversity by teaching and advising graduate students, post-doc and early career mathematicians especially those who are underrepresented in the STEM fields. The project focuses on harmonic maps in spaces with an upper curvature bound. A harmonic map between Riemannian manifolds is a solution to a certain system of elliptic partial differential equations (harmonic map equations) and is also a solution to a variational problem involving the Dirichlet energy. Since the emergence of modern geometric analysis as a core mathematical discipline, the harmonic map theory has been at the forefront of the field and important applications continue to be found. A more recent development is the study of harmonic maps into complete metric spaces satisfying an upper curvature bound. The goal of this project is to develop this theory in order to apply it to solve rigidity problems, to understand the structure of surface fibration over Kahler manifolds and projective varieties, and to study quasiconformal maps.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.