Mathematical Sciences: Geometric and Analytic Problems in Several Complex Variables and Partial Equations

DMS-9501516罗斯柴尔德/巴文迪 研究人员将研究复空间中真实的子流形的几何。特别是，他们将专注于几何和解析不变量下的坐标全纯变化这些流形。 特别考虑由真实的多项式方程全局定义的流形。 对于许多由真实的多项式的消失定义的流形，从一个这样的流形到另一个流形的全纯变换的任何分量也必须是复多项式的根。 多复变出现在世纪初作为一个自然的产物研究的职能，一个复杂的变量。 很明显，这个理论与它的前身有很大的不同。 基本的几何是更难以把握和函数理论有更多的亲和力与偏微分算子的一阶。 因此，它成长为一个混合学科结合了微分几何和微分方程的深刻特点。 许多基本结构是在过去三十年中确定的。 目前的研究仍然集中在理解这些基本的数学形式。

DMS-9501516 Rothschild/Baouendi The investigators will study the geometry of real submanifolds in complex space. In particular, they will focus on geometric and analytic invariants of these manifolds under holomorphic changes of coordinates. Special consideration will be given to those manifolds globally defined by real polynomial equations. For many manifolds defined by the vanishing of real polynomials, any component of a holomorphic transformation from one such manifold to another must also be a root of a complex polynomial. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.