The Pick–Nevanlinna problem refers to the problem of – given two connected open sets in complex Euclidean spaces and finite sets of distinct points in each – characterizing (in terms of the given point data) the existence of a holomorphic map between the two sets that interpolates the given points. The problem gets its name from Pick – who provided a beautiful characterization for the existence of an interpolant when the domain and the co-domain are the unit disc – and from Nevanlinna, who rediscovered Pick’s result. This characterization is in terms of matrix positivity. I shall begin by presenting an easy argument by Sarason, which he says is already implicit in Nevanlinna’s work, for the necessity of the Pick–Nevanlinna condition. How does one prove the sufficiency of the latter condition? Sarason’s ideas have provided the framework for a long chain of complicated characterizations for the more general problem. The last word on this is still to be written. But in the original set-up of Pick, geometry provides the answer. Surprisingly, Pick did not notice that his approach provides a very clean solution to the interpolation problem where the co-domain is any Euclidean ball. We shall see a proof of the latter. This proof uses a general observation about conditional positivity (which also made an appearance in the previous talk), which is attributed to Schur. If time permits, we shall see what can be said when the co-domain is a bounded symmetric domain.

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Last updated: 09 Apr 2019