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

多维复空间中的实曲面或子流形展示了丰富的局部几何和全局几何。真实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要素。在拟议的工作中，将使用微分和代数几何以及实数和复数分析以及偏微分方程式的技术。主要研究者计划确定两个这样的流形在可逆全纯变换下何时等价。他们将寻找新的建设性标准来建立这样的映射并确定它们的趋同。他们还计划将映射从多维复杂空间中的一个表面分类到嵌入更高维度的复杂空间中的另一个表面。特别令人感兴趣的是那些由二次方程定义并允许大对称群的超曲面。后者将作为模型，以发现更一般的现象并阐明所考虑的流形的基本性质。特别是，他们将专注于确定何时可能将双全纯等价问题归结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实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，他们将尝试对这样的映射由给定点处的有限多个导数确定的子流形进行分类。他们期望这项研究将导致发现这些子流形的新的几何、解析和代数不变量。主要研究人员将开始对一阶和更高阶的超定非线性偏微分方程组进行新的研究。他们将研究这类系统的解的性质，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引起了许多数学家和物理学家的注意，首先是亨利·庞加莱和埃利·卡尔坦的工作。到目前为止，一些根本问题仍然没有解决。对复空间中实流形几何的研究是几个复变量领域和其他科学领域的中心，包括几何、数学物理和工程学。主要调查员提出的问题所取得的进展可能也会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.