Unbounded Continuous Logic and Metric Geometry

Abstract

This thesis studies continuous logic and its application to metric geometry. An adaptation of continuous logic for unbounded pointed metric spaces is introduced and developed. Background on CAT(k) spaces, asymptotic cones, symmetric spaces, and buildings is provided. Various definability results are proved regarding geodesic rays and the building structure on them. We conclude with a proof of the instability of asymptotic cones of a certain class of symmetric spaces.