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: 1300867
• 负责人: Yuan Yuan
• 金额: $ 12.7万
• 依托单位: Johns Hopkins 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: 2013-07-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300867&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.

袁媛的这个数学研究项目涉及多个复变量和复杂微分几何中的许多问题，包括全纯结构的刚性和分类、卡勒几何中的规范度量以及复杂的蒙日-安培方程。这些是与数学和物理的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、数论、偏微分方程。特别是，袁将研究厄米对称空间上复杂结构的唯一性以及有界对称域之间的映射刚性；以及（抛物线）复数 Monge-Ampere 方程和规范卡勒度量的（有限和无限时间）极限行为之间的深层关系，以及卡勒流形上奇点的形成。复分析的数学领域占据了中心舞台十九世纪，它的应用对其他科学和工程变得至关重要，包括电子工程和机械工程。多年来，这种趋势一直持续着，事实上已经提升到了一个新的水平：袁远在这个数学研究项目中研究的几何空间可以作为宇宙学和广义相对论最基本的模型。阐明这些模型的几何结构对于理解与其相关的物理定律极其重要，并且可以帮助我们进一步理解宇宙的形状。除了这项工作外，袁还将继续参加并组织本科生、研究生和年轻研究人员的研讨会和讲习班。 袁还将指导本科生和研究生，从而使该项目有效地将研究和教育结合起来。