Mathematical Sciences: Group Definable in o-minimal Structures

Sergei Starchenko, Vanderbilt University .该项目涉及o-极小结构中可定义群的分类。虽然“o-极小结构”是模型理论的一个概念，但事实证明，实数的许多重要展开式都是o-极小的，而且这个模型理论概念为研究“解析几何”范畴提供了一个自然的框架。模型理论的工具已经成功地应用于这些范畴，得出了关于实数的有趣定理。这个项目的出发点是观察到在0 -极小结构中可定义的群可以被看作是实闭域上的李群。当只在实数领域中处理时，这些群的某些方面很容易被忽视。然而，当我们在模型-理论上表现良好的结构中将群视为可定义时，这些方面就会充分显现出来。Starchenko项目的主要目标是获得o-极小群的合理分类，以便开发李理论的模型理论模拟。他推测局部0 -极小群是代数的，李态射本质上是纳什态射。如果是这样的话，那么代数群的许多性质可以转移到0 -极小群上，从而为研究这类群提供了新的工具。由于代数实数的领域是0极小的，这些工具可以应用于在数字领域的展开中可定义的研究小组。另一方面，任何反例都将提供关于欧几里得群几何和代数性质之间相互作用的新信息。项目的第二部分涉及稳定性理论中的一些悬而未决的问题。模型理论的许多思想和工具在0 -极小结构中具有天然的几何相似物。研究者希望一个好的几何图形的存在可以帮助解决稳定性理论中两个最著名问题的0 -极小版本。该项目采用模型理论的逻辑工具，特别是称为0极小性的概念，以加深对几个重要数字系统的数学理解，包括实数。对实数及其几何性质的理解是数学的基础，也是数学的许多应用的基础，例如制造业。模型理论为这种理解提供了一个有价值的视角。

Abstract DMS 9626377 Sergei Starchenko, Vanderbilt University The project concerns the classification of groups definable in o-minimal structures. While "an o-minimal structure" is a notion of model theory, it turns out that many important expansions of real numbers are o-minimal, and this model theoretical notion provides a natural framework for the study of "analytic-geometric" categories. Tools of model theory have been successfully applied to these categories yielding interesting theorems about the real numbers. The starting point for this project is the observation that groups definable in o-minimal structures can be viewed as Lie groups over real closed fields. Certain aspects of these groups are easily overlooked when treated only over the field of real numbers. These aspects come into full view, however, when we regard the groups as definable in model-theoretically well-behaved structures. The main goal of Starchenko's project is to obtain a reasonable classification of o-minimal groups, in order to develop a model-theoretical analogue of Lie theory. He conjectures that locally o-minimal groups are algebraic and Lie morphisms are, essentially, Nash morphisms. If this is the case, then many properties of algebraic groups can be transferred to o-minimal groups, thus providing new tools to study such groups. Since the field of algebraic real numbers is o-minimal, these tools can then be applied to study groups definable in expansions of number fields. On the other hand, any negative example will provide new information about interactions between geometric and algebraic properties of Euclidean groups. The second part of the project concerns some open questions in stability theory. It has been shown that many ideas and tools of model theory have natural geometric analogues in o-minimal structures. The investigator hopes that the presence of a nice geometry can help to solve o-minimal versions of the two most famous questions in stability theory. This project employs l ogical tools of model theory, and specifically a notion called o-minimality, to deepen the mathematical understanding of several important number systems, including the real numbers. An understanding of the real numbers and of their geometric properties is basic to mathematics, and also to many applications of mathematics, e.g. to manufacturing. Model theory provides a valuable perspective in this understanding.