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

模型论是数学基础的一部分，旨在发现在许多不同的数学分支中存在的共同数学真理。经典上，它在处理一阶结构方面非常成功，例如群，线性序，布尔代数等。这个列表中缺少的一个重要概念是拓扑空间，它不能被视为一阶结构。因此，经典模型论在处理来自分析的问题时远不如处理来自代数的问题成功。模型理论的最新发展带来了新的工具，提供了弥合这一差距的潜力。特别是，一个模型理论的Banach空间已经发展。PI指出，现在是时候将这些工具应用到具体的问题，来自Banach空间理论和相关的Banach空间的conciliation.For原因中描述的建议，其中一个最有前途的领域，模型理论的方法可以使用的Banach空间理论是等距理论的各种类别的Banach空间。该提案计划匹配一个领先的专家对Banach空间的同构理论，一个领先的专家对模型理论的Banach空间和PI，其利益和专业知识进入这两个领域。该计划是有一个为期一个月的研究会议的所有三个参与者时，研究将开始和发展，其次是电子邮件合作和访问的PI到每一个其他两个参与者，讨论传播，出版和未来的研究计划。该建议可以被视为一个新的一步，在成功的计划应用无穷组合学的Banach空间理论，由高尔斯在他的菲尔兹奖工作中开始。该方案已发展成为一个国际上活跃的研究方向相结合的集合论和Banach空间理论。这项建议的新奇在于，通过纳入模型理论最新发展提供的见解，将数理逻辑的另一部分纳入混合集合论/巴拿赫群理论研究视角。希望这项资助支持的研究将形成长期互动的种子，并形成进一步研究方向的分支。

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.