Harmonic Maps into Spaces with an Upper Curvature Bound

“地图”一词的日常定义是“一个区域的陆地或海洋的图示，显示物理特征，城市，道路等。” 制图师绘制地图是为了揭示一个地理区域的有趣的空间信息。 同样，数学家在几何空间之间构造映射，以发现这些空间的有趣特征。 在这里，几何空间，我们的意思是一个空间配备的角度，距离，面积等概念，他们包括欧几里德空间，这是经常用来模拟我们的日常物理世界或非欧几里德或黎曼空间，可用于作为一个大规模的模型，我们的宇宙。 该奖项提供资金，以研究几何空间之间的特殊映射，称为调和映射，最小化某种能量概念。 通过分析调和映射，PI旨在揭示重要几何空间的性质，从而更好地理解自然世界。 调和映射的数学理论已经应用于不同的领域，如医学（例如，医学成像）和计算机科学（例如，计算机视觉），并具有进一步的潜在应用，有助于科学进步和社会福利。该项目也有一个教育组成部分，并通过教学和咨询研究生，博士后和早期职业数学家，特别是那些在STEM领域代表性不足的人来支持多样性。 该项目的重点是调和映射的曲率上界空间。黎曼流形之间的调和映射是某个椭圆型偏微分方程组（调和映射方程）的解，也是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，并不断地被发现重要的应用。最近的一个发展是研究满足曲率上界的完备度量空间中的调和映射。该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，理解Kahler流形和投影簇上的表面纤维化结构，并研究拟共形映射。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.