Complex Interpolation and Complex Convexity

斯洛德科夫斯基教授的数学研究项目题目所表示的插值概念可以解释如下。许多种类的数学对象经常不是单独出现，而是在整个家族中出现，通过指定参数的值来定位其中的单个成员。（例如，一个著名的巴拿赫空间族是由一组大于或等于1的实数来索引的，为了完整起见，还加上了正无穷。）插值的一般问题是：给定由参数的极值定位的族的成员，以某种自然的、最优的方式填充其余的成员。本项目所处理的特定插值问题对数学分析具有根本性的影响。更具体地说，这些问题与多变量复插值理论和复凸性的一些相关方面有关。它们被分为三大类。第一组问题涉及无限维空间的插值问题、正则性问题、凸集和多项式凸集的插值问题。第二类问题涉及多项式凸包。第三种是与满足类似于泛函理论性质的解析性和调和性的多函数有关。

The notion of interpolation indicated by the title of Professor Slodkowski's mathematical research project may be explained as follows. Mathematical objects of many sorts frequently come not singly but in whole families, individual members of which are located by specifying the value of a parameter. (For instance, a famous family of Banach spaces is indexed by the set of real numbers greater than or equal to one, with plus infinity thrown in for completeness.) The general problem of interpolation is: given members of the family located by extreme values of the parameter, fill in all the rest in some natural, optimal way. The particular interpolation problems addressed by this project have fundamental implications for mathematical analysis. More specifically, these problems have to do with multivariable complex interpolation theory and some related aspects of complex convexity. They are divided into three broad categories. The problems in the first group deal with the interpolation of infinite-dimensional spaces, regularity questions, and interpolation of convex and polynomially convex sets. The second category of problems concerns polynomially convex hulls. The third has to do with multifunctions that satisfy analogues of the function-theoretic properties of analyticity and harmonicity.