Mathematical Sciences: Group Definable in o-minimal Structures

摘要 DMS 9626377 谢尔盖·斯塔琴科，范德比尔特大学 该项目涉及O-最小结构中可定义的组的分类。 虽然“o-最小结构”是模型论的一个概念，但事实证明，真实的数的许多重要扩展都是o-最小的，并且这个模型论概念为“分析几何”范畴的研究提供了一个自然的框架。 模型论的工具已经成功地应用于这些范畴，产生了关于真实的数的有趣定理。 这个项目的出发点是观察到在o-极小结构中可定义的群可以被看作是真实的闭域上的李群。 当只在真实的数域上处理时，这些群的某些方面很容易被忽视。 然而，当我们把这些群看作是在模型理论上行为良好的结构中可定义的时，这些方面就完全显现出来了。 Starchenko项目的主要目标是获得o-极小群的合理分类，以开发李论的模型理论模拟。 他表示，当地O-极小群是代数和李态射，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o-极小群上，从而为研究这类群提供了新的工具。 由于该领域的代数真实的号码是O-最小的，这些工具，然后可以应用到研究组定义的扩展数域。 另一方面，任何反例都将提供关于欧几里得群的几何和代数性质之间相互作用的新信息。 本项目的第二部分涉及稳定性理论中的一些开放性问题。 模型论的许多思想和工具在o-极小结构中具有自然的几何相似性。 研究人员希望，一个好的几何的存在可以帮助解决o-最小版本的两个最著名的问题在稳定性理论。 该项目采用模型论的逻辑工具，特别是称为o-极小的概念，以加深对几个重要数字系统的数学理解，包括真实的数字。 对真实的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。 模型理论为理解这一问题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.