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

袁媛的这个数学研究项目涉及多复变和复微分几何中的一些问题，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。这些基本问题与数学和物理学的许多其他领域密切相关，如代数几何、数学物理、数论、偏微分方程等。特别是，袁将研究埃尔米特对称空间上复杂结构的唯一性和有界对称域之间的映射刚性;以及人类与自然之间（有限和无限时间）极限行为（抛物线）复Monge-Ampere方程和正则Kahler度量以及Kahler流形上奇点的形成。复分析的数学领域从Kahler流形开始占据中心舞台。世纪，当它的应用成为至关重要的其他科学和工程，包括电子工程和机械工程。多年来，这种趋势一直在持续，事实上已经达到了一个新的水平：袁媛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中最基本的模型。澄清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，并且可以帮助我们进一步理解宇宙的形状。除了这项工作，袁将继续参与，并组织研讨会和讲习班的本科生和研究生和年轻的研究人员。 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.