Topics in model theory

我建议的研究项目是模型理论，数理逻辑的一个分支，更具体地说，是稳定性和简单性理论。我目前正在进行三个广泛定义的项目：可分类理论的精细结构：当考虑在可分类理论中作为分解树上的叶子发现的模型时，会出现以下问题。设p是可数模型M上深度为零的正则类型，M(P)是由p在M上的实现所支配的模型。如果p是强正则的，则M(P)是M上的素数且是p的实现；如果p是局部模，则在与Bouscaren，Hrushovski和Laskowski的联合工作中，M(P)是众所周知的。如果p是平凡的，那么M(P)的结构目前还不被理解。这扩展了“可数理论的不可数谱”的工作。简单理论基础中的开放问题：关于简单理论的三个感兴趣的开放问题是1。消除一般简单理论中的想象，2.稳定的分叉猜想3.一般群构造和群存在定理这些问题在某种程度上与以下主题有关：一阶连续逻辑：Ben-Ya‘acov最近引入了一阶连续逻辑的概念，并开始研究各种与稳定性有关的概念。这种新设置的优点之一是，与经典的一阶情况不同，相应的超想象概念可以被认为是结构的一个真正的元素。这里有各种有趣的项目，包括在这种背景下理解经典问题的结果，确定一些经典问题在连续的上下文中是否有解决方案，并最终查看这些结果在多大程度上回到了标准设置。

My proposed research program lies in model theory, a branch of mathematical logic, and more specifically in stability and simplicity theory. I am currently working on three broadly defined projects:Fine structure for classifiable theories: The following question arises when considering models found as leaves on decomposition trees in a classifiable theory. Suppose that p is a regular type of depth zero over a countable model M and M(p) is a model dominated by a realization of p over M. If p is strongly regular than M(p) is prime over M and the realization of p; if p is locally modular then in joint work with Bouscaren, Hrushovski and Laskowski, M(p) is well-understood.  If p is trivial the structure of M(p) is currently not understood. This extends work in "The uncountable spectra of countable theories".Open problems in the foundations of simple theories: Three problems of interest that remain open regarding simple theories are1. elimination of imaginaries in a general simple theory,2. the stable forking conjecture3. general group construction and group existence theoremsThese problems are related in a way to the following topic:First-order continuous logic: Ben-Ya'acov has recently introduced the notion of first-order continuous logic and has begun studying various stability related notions.  One of the advantages of this new setting is that the corresponding notion of hyperimaginary can be thought of as a "real" element of the structure, unlike the classical first order case.  There are a variety of projects here that are interesting including understanding the outcome of classical problems in this setting, determining if some of the classical problems have solutions in the continuous context and finally seeing to what degree these results pull back to the standard setting.