Geometric and Analytic Problems in Several Complex Variables

确定多维复空间中的两个真实的子流形何时局部双全纯等价是多复变几何与分析中的基本问题之一。也就是说，什么时候有可能找到一个局部可逆的全纯变换将一个子流形发送到另一个子流形上？自世纪以来，这个问题引起了许多数学家的注意，从庞加莱的工作开始，一直到E.嘉当、田中、陈省身、莫泽等人。主要研究人员将继续对这个问题的几个方面进行研究。特别是，他们将专注于确定何时可以将双全纯等价问题简化为求解复系数多项式方程组。他们还计划确定将一个真实的子流形发送到另一个子流形的形式映射何时必然收敛。此外，他们将试图对那些由给定点上的许多导数确定的映射的子流形进行分类。他们希望这项研究将导致发现新的几何，分析，以及代数不变量的这些submanifold.The主要研究人员将继续他们的基本性质的分析和几何对象，如曲面和曲线，在多维复杂的空间。这些数学对象的完整分类对数学科学中的许多其他问题有重要的意义。 事实上，这项研究的动机是数学和物理学的几个领域之间丰富的相互作用，包括控制论，弦理论和数学物理学的其他领域。在主要调查员提出的问题上取得进展也可能对上述领域产生影响。

One of the basic geometric and analytic problems in several complex variables is to determine when two real submanifolds in multidimensional complex space are locally biholomorphically equivalent. That is, when is it possible to find a local invertible holomorphic transformation sending one submanifold onto the other? This problem has attracted the attention of many mathematicians since the beginning of the twentieth century, starting with the work of Poincare and continuing with the major contributions of E. Cartan, Tanaka, Chern, Moser and others. The principal investigators will continue their research on several aspects of this problem. In particular, they will focus on determining when it is possible to reduce the biholomorphic equivalence problem to solving systems of polynomial equations with complex coefficients. They also plan to determine when a formal mapping sending a real submanifold into another is necessarily convergent. In addition, they will attempt to categorize those submanifolds for which such mappings are determined by finitely many derivatives at a given point. They expect that this study will lead to the discovery of new geometric, analytic, as well as algebraic invariants of these submanifolds.The Principal Investigators will continue their study of fundamental properties of analytic and geometric objects, such as surfaces and curves, in multidimensional complex spaces. A complete classification of these mathematical objects can have important implications for a number of other questions in mathematical science. In fact, this study is motivated by the rich interplay between several areas of mathematics and physics, including control theory, string theory, and other areas of mathematical physics. Progress on the problems proposed by the Principal Investigators will likely have impact on the above-mentioned areas as well.