Mathematical Sciences: Analytic Discs and Integral Formulas in Several Complex Variables

图曼诺夫9401652该奖项支持对多复变函数理论中出现的问题进行数学研究。这项工作集中在全纯函数的边值上；更准确地说，是识别区域边界上的函数，这些函数可以是全纯函数的边值。这样的函数必须满足称为边界柯西-黎曼方程的偏微分方程组。解决方案称为CR函数。用于从流形扩展CR-函数的技术包括一种新发展的过程，即将圆盘附加到流形上，并通过圆盘查看解析扩张以获得全纯扩张。然后，必须将圆盘和延伸部适当地缝合在一起。另一种方法涉及积分公式，它通过使用依赖于流形的核(Szego核)的积分过程给出扩展。还将对一个长期存在的问题进行新的研究，即在多个复变量域中嵌入CR结构。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。***

Tumanov 9401652 This award supports mathematical research on problems arising in the theory of functions of several complex variables. The work concentrates on boundary values of holomorphic functions; more precisely that of identifying functions on the boundary of a domain which can be boundary values of holomorphic functions. Such functions must satisfy partial differential equations known as the boundary Cauchy-Riemann equations. Solutions are known as CR-functions. Techniques for extending CR-functions from manifolds include a newly developing process of attaching discs to manifolds and looking at analytic extensions through the discs to obtain holomorphic extension. Then the discs and extensions must be sewn together appropriately. Another method involves integral formulas which give the extension by means of an integration process using a kernel which depends on the manifold (Szego kernel). A new line of research on a long-standing problem of embedding CR-structures in domains of several complex variables will also be undertaken. 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. ***