Research in Model Theory

这项工作涉及到稳定性理论，代数模型理论和有限模型理论的研究的统一。 一个基本的技术是同构的通用模型的建设方面的一个概念，强子模型。 一方面，这将稳定性理论与有限模型上的概率联系起来，不仅提供了证明0-1定律的技术，而且还提供了几乎必然理论的模型论性质。 几乎处处理论的这些模型论性质可以应用于有限模型理论中的问题。 另一方面，这种技术被用来试图构建一个坏场。 最后，嵌入式有限模型的研究解决了数据库数据库理论的问题，使用稳定性理论的技术。 在这项工作的一个主题是模型理论，一个分支的数理逻辑。 总的目的是在更高的抽象层次上理解“普通数学”。 这以前导致在代数领域的应用，特别是群论。 本项目将尝试寻找一些特定的新结构来扩展复数。 其中一些相同的见解将用于探索有限图的渐近性质。 这个抽象的框架的进一步应用已经在“嵌入式有限模型理论”的发展，其中的模型理论的技术被用来找到数据库查询的可表达性的限制。

This work involves a uniting of investigations in stability theory, model theory of algebra, and finite model theory. One underlying technique is the construction of homogeneous universal models with respect to a notion of strong submodel. On the one hand, this links stability theory with probability on finite models by providing a technique for not only proving 0-1 laws but obtaining model theoretic properties of the almost sure theory. These model theoretic properties of the almost sure theory can be applied to problems in finite model theory. On the other hand, this technique is exploited to attempt to construct a bad field. Finally, investigations of embedded finite models solves problems of database database theory using techniques from stability theory. One theme in this work is model theory, a branch of mathematical logic. The general aim is to understand `ordinary mathematics' at a higher level of abstraction. This has previously resulted in applications in the area of algebra, specifically group theory. The present project will involve an attempt to find some specific new structures which expand the complex numbers. Some of these same insights will be used to explore the asymptotic properties of finite graphs. A further application of this abstract framework has been in the development of `embedded finite model theory,' where the techniques of model theory are employed to find limits on the expressibility of database queries.