Problems on the geometric function theory in several complex variables and complex geometry

几何函数论中的多复变数和复几何问题

• 批准号: 1412384
• 负责人: Yuan Yuan
• 金额: $ 12.7万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1412384&HistoricalAwards=false
• 关键词: Problems; geometric; function; theory; several

This mathematics research project by Yuan Yuan concerns a number of problems in several complex variables and complex differential geometry, consisting of the rigidity and classification of holomorphic structures, canonical metrics in Kahler geometry, and complex Monge-Ampere equations. These are fundamental problems closely related to many other fields in mathematics and physics, such as, algebraic geometry, mathematical physics, number theory, partial differential equations. In particular, Yuan will study the uniqueness of complex structure on Hermitian symmetric spaces and mapping rigidity between bounded symmetric domains; and the deep relation between the (finite and infinite time) limit behavior of the (parabolic) complex Monge-Ampere equations and canonical Kahler metrics as well as the formation of singularities on Kahler manifolds.The mathematics field of complex analysis took center stage starting with the nineteenth century, when its applications became crucial to other sciences and engineering, including electronic engineering and mechanic engineering. Over the years, this trend has continued and in fact has been taken to the next level: the geometric spaces studied in this mathematics research project by Yuan Yuan can serve as the most basic models in cosmology and general relativity. Clarifying the geometric structure of these models is extremely important in understanding the physical laws that relate to them and can help further our understanding of the shape of the universe. In addition to this work, Yuan will continue to participate in, and organize seminars and workshops for undergraduate and graduate students and young researchers. Yuan will also mentor undergraduate and graduate students, and in this way the project will effectively integrate research and education.

元元的该数学研究项目涉及多个复杂变量和复杂的差异几何形状的许多问题，包括全体形态结构的刚性和分类，Kahler几何形状的规范指标以及复杂的Monge-Monge-Ampere方程。这些是与数学和物理学中许多其他领域密切相关的基本问题，例如代数几何，数学物理学，数量理论，部分微分方程。特别是，人民币将研究遗体对称空间上复杂结构的独特性，并在有界对称结构域之间绘制刚性。以及（抛物线和无限时间）的（抛物线）复杂的蒙格 - 安培方程和规范的卡勒指标（Kahler指标）以及卡勒（Kahler）在卡勒（Kahler）歧管上的奇异现象的形成之间的深厚关系。多年来，这种趋势一直在继续，实际上已经提升到了一个新的水平：元元基于该数学研究项目所研究的几何空间可以作为宇宙学和一般相对论中最基本的模型。阐明这些模型的几何结构对于理解与它们相关的物理定律非常重要，并可以进一步帮助我们理解宇宙的形状。除了这项工作外，元还将继续参加，并为本科生和研究生和年轻研究人员组织研讨会和研讨会。 Yuan还将指导本科生和研究生，这样该项目将有效地整合研究和教育。