Mathematics Colloquia and Seminars

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\par Consider the following problem. Suppose that you are given points $z_1, \dots, z_n$ in the unit disk $\mathbb{D}$ of the complex plane and complex numbers $w_1, \dots, w_n.$ Does there exist a holomorphic function $\phi: \mathbb{D}\to\mathbb{D}$ such that \[\phi(z_i)=w_i, \ \  i=1, \dots, n\text{?}\] Pick's solution of this problem in 1915 impacted the development of function theory throughout the twentieth century. In 1967, Sarason gave an operator-theoretic reformulation of Pick's result and proved his seminal commutant lifting theorem, which was subsequently generalized by Sz.-Nagy and Foiaș into a powerful tool that unifies and extends a variety of classical interpolation and moment theorems on $\mathbb{D}$.   \par In this talk, I will discuss Sarason's approach to the Pick problem and how it connects to a certain Hilbert space of holomorphic functions. Further, I will explain how Sarason's theorem can be extended to more general spaces. This is ongoing joint work with Scott McCullough (University of Florida).