The Kim-Pillay theorem for Abstract Elementary Categories

Normal hyperimaginaries

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of Lascar-Pillay's theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of Peter-Weyl's theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of $φ$-types for bounded hyperimaginaries.

Kim-independence in positive logic

An important dividing line in the class of unstable theories is being NSOP$_1$, which is more general than being simple. In NSOP$_1$ theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP$_1$ theories to positive logic -- a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get anoth…

Ramsey theory and topological dynamics for first order theories

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide cla…

论一阶顺从性

On first order amenability

本文引入一阶理论的一阶顺从性概念，研究其基本性质与等价条件，证明顺从理论是G-紧的，推广了ω-范畴理论的相关结论，还得到极端顺从性的平行结果。

近似等价关系

Approximate equivalence relations

本文在可定义测度下研究近似等价关系，基于连续逻辑提出概率逻辑框架，利用Kim-Pillay空间完全分类二元相关性，推广范畴性定理与近似子群的李模型定理，并证明NIP理论中可定义测度自同构群的阿基米德有限维性。

新稳定性理论中的抽象独立关系

Abstract independence relations in neostability theory

本文在Adler框架下发展抽象独立关系间的“见证”概念，统一新稳定性理论中的Kim引理变体，证明见证与对称性等价，并推导出链局部特征、弱独立定理等结果，最后证明NSOP₁与BTP之间的二分法。