Interpolation and Related Topics

Washington University, Saint Louis MO

Investigators

Abstract

Abstract McCarthy Pick interpolation has, for the last 30 years, been a very successful blend of functional analysis, operator theory and function theory. The proposer intends to study it in the context of complex geometry. The more specific goals are to understand interpolating sequences in many algebras - not just the algebra of bounded analytic functions on some domain, but multiplier algebras of more general function spaces. More generally, viewing interpolation problems as questions about curvature, the proposer seeks to understand the geometry of the maximal ideal spaces. Feedback mechanisms have been used in engineering for a long time to stabilize systems. The design of such systems is difficult, not least because the system that one wants to stabilize is, in practice, not known exactly, but only approximately. Many ad hoc approaches were used, but about 20 years ago an idea surfaced that enabled the problem to be confronted directly. This was a form of control theory that allowed robust stabilization. It relied on a mathematical technique called interpolation - finding a nice function that takes certain values. The aim of the current proposal is to greatly increase the class of interpolation problems that can be solved, so that systems that must operate under two qualitatively different constraints simultaneously can be stabilized.