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

原媛的这项数学研究项目涉及多个复变量和复微分几何中的一系列问题，包括全纯结构的刚性和分类，Kahler几何中的正则度量，以及复Monge-Ampere方程。这些都是与数学和物理中的许多其他领域密切相关的基本问题，如代数几何、数学物理、数论、偏微分方程组。特别是，袁将研究厄米特对称空间上复结构的唯一性和有界对称域之间的映射刚性；(抛物型)复Monge-Ampere方程的(有限和无限时间)极限行为与正则Kahler度量以及Kahler流形上奇点形成之间的深层联系。复分析的数学领域从19世纪开始成为中心，当时它的应用对其他科学和工程，包括电子工程和机械工程变得至关重要。多年来，这一趋势一直在继续，事实上已经更上一层楼：原媛在这项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中最基本的模型。弄清这些模型的几何结构对于理解与它们相关的物理规律极其重要，并有助于进一步了解宇宙的形状。除了这项工作，袁还将继续参与并为本科生、研究生和青年研究人员组织研讨会和研讨会。袁还将指导本科生和研究生，通过这种方式，该项目将有效地整合研究和教育。

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.