Harmonic Maps and Their Applications

谐波图及其应用

• 批准号: 1709475
• 负责人: Chikako Mese
• 金额: $ 24.65万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709475&HistoricalAwards=false
• 关键词: Harmonic; Maps; Their; Applications

Most objects in nature are not smooth. Yet most existing literature in geometry deal with smooth geometric objects such as a plane, a sphere, a saddle surface and their generalizations. In this NSF funded project, the PI proposes to study singular geometry by analyzing maps into and between them that are optimal in the sense that they minimize energy. The analysis of these maps will help us understand symmetry properties that exists in certain singular spaces. The project will advance our scientific knowledge in the understanding of non-singular geometry; in particular, the PI's research will lead to a greater understanding of the natural world. The project also has an education component and supports diversity by teaching and advising graduate students, post-doc and early career mathematicians especially those that are underrepresented in the STEM fields. A natural notion of energy for a map between geometric spaces is defined by measuring the total stretch of the map at each point of the domain and then integrating. Harmonic maps are critical points of the energy functional. They can be seen as both a generalization of harmonic functions in complex analysis and a higher dimensional analogue of parametrized geodesics in Riemannian geometry. Next to totally geodesic maps, harmonic maps are perhaps the most natural way to map a given geometric space into another. The celebrated work of Eells and Sampson launched an explosion of research in harmonic maps between Riemannian manifolds. Many important applications were found, particularly in minimal surface theory, Teichmuller theory and rigidity questions. A more recent development is the harmonic map theory for non-smooth spaces. The seminal works of Gromov-Schoen and Korevaar-Schoen study energy minimizing maps in the case when the target is an non-positively curved (NPC) space. These energy minimizing maps are referred to as harmonic maps. The PI will extend this harmonic map theory for singular geometry and to apply it to solve problems in other fields.

自然界中的大多数物体都不是光滑的。 然而，大多数现有的文献中的几何处理光滑的几何对象，如一个平面，一个球，一个鞍曲面及其推广。 在这个NSF资助的项目中，PI建议通过分析它们之间的映射来研究奇异几何，这些映射在最小化能量的意义上是最优的。 对这些映射的分析将帮助我们理解存在于某些奇异空间中的对称性。 该项目将推进我们对非奇异几何的理解;特别是PI的研究将导致对自然世界的更深入了解。 该项目还包括教育部分，通过为研究生、博士后和早期职业数学家提供教学和咨询，特别是那些在STEM领域代表性不足的数学家，支持多样性。 几何空间之间映射的能量的一个自然概念是通过测量映射在域的每个点处的总拉伸然后积分来定义的。 调和映射是能量泛函的临界点。 它们既可以被看作是复分析中调和函数的推广，也可以被看作是黎曼几何中参数化测地线的高维模拟。 除了全测地映射，调和映射可能是将给定几何空间映射到另一个几何空间的最自然的方法。 著名的工作埃尔斯和桑普森发起了爆炸的研究调和映射之间的黎曼流形。 许多重要的应用发现，特别是在极小曲面理论，Teichmuller理论和刚性问题。 最近的发展是非光滑空间的调和映射理论。 Gromov-Schoen和Korevaar-Schoen的开创性工作研究了当目标是非正弯曲（NPC）空间时的能量最小化映射。这些能量最小化映射被称为调和映射。 PI将扩展这一调和映射理论的奇异几何，并将其应用于解决其他领域的问题。

项目成果

Regularity of harmonic maps from polyhedra to CAT(1) spaces

• DOI: 10.1007/s00526-017-1279-5
• 发表时间: 2016-10
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Christine Breiner;A. Fraser;Lan-Hsuan Huang;Chikako Mese;Pam Sargent;Yingying Zhang

Higgs bundles over cell complexes and representations of finitely presented groups

细胞复合体上的希格斯丛和有限呈现群的表示

• DOI: 10.2140/pjm.2018.296.31
• 发表时间: 2018
• 期刊: Pacific Journal of Mathematics
• 影响因子: 0.6
• 作者: Daskalopoulos, Georgios;Mese, Chikako;Wilkin, Graeme
• 通讯作者: Wilkin, Graeme