Prerequisites: Math 776 optional
Time and Place: MWF 12:05 - 12:55
Course Description: Recently, model theory has become a rich area of research with many interactions to other fields: algebra, combinatorics, number theory, etc. Yet until recently, the inherrently discrete nature of model-theoretic analysis prevented model theory from offering these same tools to more continuous structures. Thus enters continuous model theory. In continuous model theory, rather than having a Boolean truth value (true or false), every statement can be given a truth value in a bounded interval in the reals (say, like a probability of an event in a probability space or a distance in a metric space).
In the first part of the semester, we will develop the general theory of continuous model theory. Though much is done analogously to standard first-order logic, our development will (necessarily) be from the foundation, and thus this course can be taken without having taking Math 776.
In the latter part of the semester, we will look at particular theories and discuss applications of continuous model theory to the study of Hilbert spaces, probability algebras, Banach lattices, ergodic theory, Keisler randomizations, the Urysohn sphere, and perhaps others. Given time, we will also touch on some recent work in the theory of operator algebras from the perspective of continuous logic.