Stability Theory in Continuous First Order Logic

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First order logic is suitable for describing and studying classes of"discrete" structures: fields, groups, graphs, etc. Model theory, andin particular stability theory, provided several striking structuretheorems for such classes: for example, the fact that every vectorspace is determined by (the cardinality of) a linear base, while everyalgebraically closed fields is determined by a transcendence base, areboth special cases of Morley's Theorem: in any class of structure towhich this theorem applies, all structures are generated by a suitablebase. Shelah's classification theory is a vast generalisation ofMorley's Theorem, yielding structure theorems to many more classes ofstructures. While all these theories were developed for classes ofdiscrete structures, they seems to hold, at least to some extent, forclasses of metric, or "continuous" structures: for example, compareMorley's theorem with the fact that every Hilbert space is generated(as a complete metric vector space) by an orthonormal base. Thisproposal seeks to further adapt classical structure results fromstability and classification theory to classes of continuousstructures.This proposal therefore seeks to extend the results and techniques ofstability theory, and in particular superstability, to the setting ofcontinuous first order logic, aiming towards a generalisation ofShelah's Main Gap Theorem. Continuous first order logic has theadvantage 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 metricrequire us to revise the fundamental definitions (superstability,ranks, etc.), and renders most of the classical theory, and inparticular the notion of regular types, seemingly inapplicable.Nevertheless there has been considerable progress recently in thisdirection, such as an adaptation of Lachlan's theorem to continuoussuperstable theories, and there are indications that similartechniques can be used towards finiteness of weight and itsconsequences, which seem the natural next step towards the achievementof this programme.