Uncountable categoricity of structures based on Banach spaces

A continuous theory T of bounded metric structures is said to be kappa-categorical if T has a unique model of density kappa. Work of Ben Yaacov and Shelah+Usvyatsov shows that Morley’s Theorem holds in this context: if T has a countable signature and is kappa-categorical for some uncountable kappa, then T is kappa-categorical for all uncountable kappa. In classical (discrete) model theory, there are several characterizations of uncountable categoricity. For example, there is a structure theorem for uncountably categorical theories T, due to Baldwin+Lachlan: there is a strongly minimal set D defined over the prime model of T such that every uncountable model M of T is minimal and prime over D(M). Moreover (and easier), if T has such a strongly minimal set, then T is uncountably categorical.In the more general metric structure setting, nothing remotely like this is known. Indeed, the metric analog of a strongly minimal set is nowhere to be seen, at the moment. If one restricts attention to metric structures based on (unit balls) of Banach structures, more is known. The appropriate analog of strongly minimal sets seems to be the unit balls of Hilbert spaces. After the speaker called attention to this phenomenon in some examples from functional analysis, Shelah and Usvyatsov investigated it and proved a remarkable result (arxiv 1402.6513