Geometric and Analytic Problems in Several Complex Variables and Partial Differential Equations

多维复空间中的真实的曲面或子流形表现出丰富的局部几何和整体几何。 真实的和复杂的几何和分析之间的相互作用是研究这些对象的基本成分。 技术从微分和代数几何，以及真实的和复杂的分析和偏微分方程，将被用于拟议的工作。主要研究人员计划确定两个这样的流形在可逆全纯变换下何时是等价的。他们将寻找新的建设性标准来建立这种映射并确定其收敛性。他们还计划对从多维复杂空间中的一个表面到嵌入更高维复杂空间中的另一个表面的映射进行分类。特别令人感兴趣的是那些由二次方程定义的超曲面，并允许大的对称群。 后者将作为模型来发现更普遍的现象，并制定考虑中的流形的基本性质。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，分析和代数不变量的这些子流形。主要研究人员将开始对一阶和高阶非线性偏微分方程的超定系统进行新的研究。他们将研究这些系统的解的性质，以便用规定的柯西数据对所有可能的子流形进行分类。 自20世纪初以来，这里讨论的几个问题吸引了许多数学家和物理学家的注意力，从亨利·庞加莱和埃利·嘉当的工作开始。一些基本问题至今仍未解决。复空间中真实的流形的几何研究是多复变领域和其他科学领域的核心，包括几何学、数学物理学和工程学。 在主要调查人员提出的问题上取得的进展也可能对这些领域产生影响。

Real surfaces or submanifolds in multidimensional complex spaces exhibit a rich local as well as global geometry. The interplay between real and complex geometry and analysis is a fundamental ingredient in studying these objects. Techniques from differential and algebraic geometry, as well as real and complex analysis and partial differential equations, will be used in the proposed work. The principal investigators plan to determine when two such manifolds are equivalent under invertible holomorphic transformations. They will look for new constructive criteria to build such mappings and to determine their convergence. They also plan to classify mappings from one surface in a multidimensional complex space into another embedded in a complex space of higher dimension. Of particular interest are those hypersurfaces defined by quadratic equations and admitting large symmetry groups. The latter will serve as models to discover more general phenomena and formulate basic properties of the manifolds under consideration. 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, and algebraic invariants of these submanifolds. The principal investigators will initiate new studies of overdetermined systems of nonlinear partial differential equations of first and higher order. They will study the properties of solutions of such systems in order to classify all possible submanifolds with prescribed Cauchy data. Several problems discussed here have attracted the attention of many mathematicians and physicists since the beginning of the twentieth century, starting with the work of Henri Poincare and Elie Cartan. A number of fundamental problems remain unsolved to the present time. The study of the geometry of real manifolds in complex spaces is central to the field of several complex variables and to other areas of science, including geometry, mathematical physics, and engineering. Progress on the problems proposed by the principal investigators will likely have impact on these areas as well.