Complex Interpolation and Complex Convexity

斯洛德科夫斯基教授的数学研究项目的标题所指出的插值法的概念可以解释如下。许多种类的数学对象经常不是单独出现的，而是整个家庭中的，家庭的单个成员通过指定参数的值来定位。(例如，一个著名的Banach空间族通过大于或等于1的实数集来索引，并加上无穷大以保证完整性。)插值法的一般问题是：给定由参数的极值确定的家族成员，以某种自然的、最佳的方式填充所有其余的参数。本项目所涉及的特殊内插问题对数学分析具有基本意义。更具体地说，这些问题与多变量复数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内插，正则性问题，以及凸集和多项式凸集的内插。第二类问题涉及多项式凸壳。第三个问题涉及满足类似函数理论性质的解析性和和谐性的多功能。

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.