First order logic is suitable for describing and studying classes of "discrete" structures: fields, groups, graphs, etc. Model theory, and in particular stability theory, provided several striking structure theorems for such classes: for example, the fact that every vector space is determined by (the cardinality of) a linear base, while every algebraically closed fields is determined by a transcendence base, are both special cases of Morley's Theorem: in any class of structure to which this theorem applies, all structures are generated by a suitable base. Shelah's classification theory is a vast generalisation of Morley's Theorem, yielding structure theorems to many more classes of structures. While all these theories were developed for classes of discrete structures, they seems to hold, at least to some extent, for classes of metric, or "continuous" structures: for example, compare Morley's theorem with the fact that every Hilbert space is generated (as a complete metric vector space) by an orthonormal base. This proposal seeks to further adapt classical structure results from stability and classification theory to classes of continuous structures.
This proposal therefore seeks to extend the results and techniques of stability theory, and in particular superstability, to the setting of continuous first order logic, aiming towards a generalisation of Shelah's Main Gap Theorem. Continuous first order logic has the advantage of being a natural generalisation of first order logic, while at the time accommodating many natural classes of (metric) structures arising in functional analysis and probability theory (various classes of Banach spaces possibly with additional structure, measure algebras of probability spaces and of adapted spaces, etc.) New complications arising from the presence of a non-discrete metric require us to revise the fundamental definitions (superstability, ranks, etc.), and renders most of the classical theory, and in particular the notion of regular types, seemingly inapplicable. Nevertheless there has been considerable progress recently in this direction, such as an adaptation of Lachlan's theorem to continuous superstable theories, and there are indications that similar techniques can be used towards finiteness of weight and its consequences, which seem the natural next step towards the achievement of this programme.
