We introduce piecewise interpretable Hilbert spaces and show their relevance to model theory and representation theory. Piecewise interpretable Hilbert spaces are direct limits of imaginary sorts of a continuous logic structure which carry definable Hilbert space operations. We will show that they offer an interesting unified framework for studying definable measures and Shelah-Galois groups, and that they offer an interesting point of contact between model theory and the theory of unitary group representations. We will briefly discuss a structure theorem for scattered piecewise interpretable Hilbert spaces and we will explain various applications of this theorem. This is joint work with Ehud Hrushovski.
In this talk we will focus on giving an overview of results and on discussing examples. More detailed results will be discussed on Tuesday in the Théorie des Modèles et Groupes seminar. | 
