Definable groups in d-minimal and related structures

这个项目是模型论(数理逻辑)和群论的结合点。主要的研究对象是可定义在各种结构中的群，这些结构可以是拓扑/几何性质的，例如o-极小结构和它们的温和扩展，例如d-极小结构。在o-极小的背景下，可定义的群已经被很好地理解了，也许最显著的结果是Pillay猜想的解，它在这些群和实李群之间建立了明确的联系。本项目将研究Pillay猜想在o-极小结构的许多可能展开式之一中的推广。具体的例子包括：(A)通过对所有2的整数幂的集合的谓词来扩展实域，(B)通过对于整数集合的谓词来扩展实序群。这些设置最近看到了模型理论工具的发展，将在本项目中使用。潜在的应用包括：-在上述设置中的紧支配猜想的解-在上述设置中的可定义群的仿射嵌入。

This project lies at the nexus of model theory (mathematical logic) and group theory. The main objects of study are groups definable in various structures, which can be of topological/geometric nature, such as o-minimal structures and tame expansions of them, such as d-minimal structures.In the o-minimal setting, definable groups have been fairly understood with perhaps the most notable result being the solution of Pillay's conjecture, which draws an explicit connection between those groups and real Lie groups. Extensions of Pillay's conjecture in one of many possible expansions of o-minimal structures will be investigated in this project. Concrete examples include(a) the expansion of the real field by a predicate for the set of all integer powers of 2,(b) the expansion of the real ordered group by a predicate for the set of integer numbersThose settings have recently seen the development of model-theoretic tools, which will be used in this project.Potential applications include:- The solution of Compact Domination Conjecture in the above settings- Affine embeddings for definable groups in the above settings.