Geometric and Analytic Problems in Several Complex Variables

多个复杂变量的几何和解析问题

• 批准号: 0100330
• 负责人: Linda Rothschild
• 金额: $ 20.2万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100330&HistoricalAwards=false
• 关键词: Geometric; Analytic; Problems; Several; Complex

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.

多个复变量中的基本几何和解析问题之一是确定多维复空间中的两个实子流形何时局部双全纯等价。也就是说，什么时候可以找到将一个子流形发送到另一个子流形的局部可逆全纯变换？自二十世纪初以来，这个问题吸引了许多数学家的关注，从庞加莱的工作开始，一直到嘉当、田中、陈省身、莫泽等人的主要贡献。主要研究人员将继续研究这个问题的几个方面。特别是，他们将重点关注确定何时可以将双全纯等价问题简化为求解具有复系数的多项式方程组。他们还计划确定将一个真实子流形发送到另一个子流形的正式映射何时必然收敛。此外，他们将尝试对那些子流形进行分类，这些子流形的映射是由给定点的有限多个导数确定的。他们期望这项研究将导致发现这些子流形的新几何、解析以及代数不变量。主要研究人员将继续研究多维复杂中的解析和几何对象的基本属性，例如曲面和曲线空间。这些数学对象的完整分类可以对数学科学中的许多其他问题产生重要影响。 事实上，这项研究的动机是数学和物理的多个领域之间丰富的相互作用，包括控制理论、弦理论和数学物理的其他领域。首席研究员提出的问题的进展也可能对上述领域产生影响。