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

DMS-9501516 Rothschild/Baouendi研究人员将研究复空间中实子流形的几何。特别是，他们将专注于这些流形在全纯坐标变化下的几何和解析不变量。我们将特别考虑那些由实多项式方程全局定义的流形。对于许多由实多项式的消失定义的流形，从一个这样的流形到另一个这样的流形的全纯变换的任何分量也一定是复多项式的根。本世纪初出现了几个复变量，这是一个复变量函数研究的自然结果。很早就很明显，这一理论与其前身有很大不同。基本的几何学要难得多，而函数论与一阶偏微算符的亲和力要强得多。因此，它成长为一门混合学科，结合了微分几何和微分方程式的深刻特征。许多基本结构都是在过去30年里定义的。目前的研究仍然集中在理解这些基本的数学形式上。

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.