Complex Interpolation and Complex Convexity

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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.