Interactions of model theory and set theory with Banach space theory; isometric structure of Banach spaces

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Model theory is part of mathematical foundations that aims to discover common mathematical truths that hold in many different branches of mathematics.Classicaly it has been very successful in dealing with first order structures, such as groups, linear orders, Boolean algebras etc. An important notion missingon this list is that of a topological space, which cannot be seen as a first order structure. Therefore classical model theory is much less successful indealing with problems coming from analysis than those coming from algebra. Recent developments in model theory have brought about new tools that offerpotential to bridge this gap. In particular, a model theory of Banach spaces have been developed.The PI makes the point that it is now time to apply these tools to specific problems coming from Banach space theory and relevant to the Banach spacecommunity.For reasons described in the proposal, one of the most promising areas of Banach space theory where model theoretic methods could be used isthe isometric theory of various classes of Banach spaces. The proposal plans to match a leading expert on the isomorphic theory of Banach spaces,a leading expert on the model theory of Banach spaces and the PI, whose interests and expertise go into both of these areas. The plan is to havea one month long research meeting of all three participants when the research would be started and developed, followed by email collaborationand a visit by the PI to each of the other two participants to discuss dissemination, publication and future research plans.The proposal can be seen as a new step in the successful programme of applications of infinitary combinatorics to Banach space theory, asstarted by Gowers in his Fields Medal work. That programme has grown into an internationally active research direction combining set theoryand Banach space theory. The novelty of this proposal is that another part of mathematical logic is included in this mixed set theory/Banachgroup theory research perspective, by including the insights offered by recent top developments in model theory.It is hoped that the research supported by this grant will form a seed of a long term interaction and branch into further research directions.