Geometric and Analytic Problems in Several Complex Variables

摘要：Rothschild和Baouendia多复变空间中的一个基本几何和解析问题是确定多维复空间中的两个真实的流形在解析变换下何时等价。主要研究人员将继续对这一基本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等价问题简化为求解多项式方程组的简单得多的问题。他们还将研究这些流形之间的映射是由给定点的许多导数决定的。他们希望这项研究将导致发现这些重要几何对象的新的几何，分析和代数性质。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。在所提出的问题上取得的进展也可能对这些领域产生影响。

ABSTRACT:Rothschild and BaouendiA basic geometric and analytic problem in several complex variables is to determine when two real manifolds in multidimensional complex space are equivalent under an analytic transformation. The principal investigators will continue their research on several aspects of this fundamental problem. In particular, they will focus on determining when it is possible to reduce the equivalence problem to the much simpler one of solving systems of polynomial equations. They will also study which mappings between such manifolds 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 properties of these important geometric objects. 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 will likely have impact on these areas as well.