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

模型理论是数学基础的一部分，旨在发现存在于许多不同数学分支中的共同数学真理。经典地，它在处理一阶结构方面非常成功，如群、线性阶、布尔代数等。在这一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，经典模型理论对来自分析的问题的索引远不如那些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，为弥合这一差距提供了潜力。特别地，发展了Banach空间的模型理论。PI指出，现在是时候将这些工具应用于来自Banach空间理论的与Banach空间社区相关的特定问题。由于提案中描述的原因，可以使用模型理论方法的Banach空间理论中最有前途的领域之一是各类Banach空间的等距理论。该提案计划匹配一位在Banach空间同构理论方面的领先专家，一位在Banach空间模型理论和PI方面的领先专家，他的兴趣和专业知识都在这两个领域。该计划将在研究开始和发展时，由所有三名参与者举行为期一个月的研究会议，随后通过电子邮件合作，并由PI访问其他两名参与者，讨论传播、出版和未来的研究计划。这项提议可以被视为高尔斯在他的菲尔德奖章工作中启动的将无限组合学应用于Banach空间理论的成功计划的新步骤。该方案已发展成为集集合论和Banach空间理论于一体的国际上活跃的研究方向。这项建议的新颖之处在于，通过将模型理论的最新发展提供的见解纳入到这种混合集合论/Banachgroup理论的研究视角中，数理逻辑的另一部分被纳入其中。希望这项拨款支持的研究将形成长期互动的种子，并分支到进一步的研究方向。

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.