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

Tumanov 9401652 该奖项支持数学研究中出现的问题的理论函数的几个复杂的变量。 工作集中在边界值的全纯函数，更确切地说，确定功能的边界上的一个域，可以边界值的全纯函数。 这样的函数必须满足被称为边界柯西-黎曼方程的偏微分方程。 解称为CR函数。 从流形扩展CR-函数的技术包括一个新发展的过程，将圆盘附加到流形上，并通过圆盘查看解析扩展以获得全纯扩展。 然后，光盘和扩展必须缝在一起适当。另一种方法涉及积分公式，该积分公式通过使用依赖于流形的核（Szego核）的积分过程来给出扩展。一个新的研究线上的一个长期存在的问题，嵌入CR-结构域的几个复杂的变量也将进行。 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掌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。 ***

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. ***