Résume  A continuous theory T of bounded metric structures is said to be kappacategorical 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 kappacategorical for some uncountable kappa, then T is kappacategorical 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; to appear in Adv. in Math.): if M is a nonseparable Banach structure (with countable signature) whose theory is uncountably categorical, then M is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of M. There is a wide gap between this result and what is true of verified examples of uncountably categorical Banach structures , which leads to the question: can a stronger such result be proved, which gives a characterization of uncountable categoricity for Banach structures and in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis?
In addition to the above background, we will discuss some new examples of uncountably categorical Banach spaces (of which there have been very few previously known). This is joint work with Yves Raynaud (Paris 6); we have a 2016 paper in Comment. Math. (now freely available on their website) and the examples to be discussed here are more recent.
