Mathematical Sciences: Set Theory and Analysis

Carlson将致力于以下四个问题及相关问题：(1)强稀疏集合形成理想吗？(2)强覆盖引理在多大程度上沿着内模型层次成立？(3)Frisse关于具有许多同构子结构的模型的猜想是否一致错误？(4)在集合论中能证明中间极限的存在吗？这些是公理集合论的基础上的技术问题。人们想知道它们在什么假设下是正确的，是否存在它们失败的集合论模型和它们成立的其他集合论模型。正是这样的微妙问题让我们能够探索所有数学逻辑所依赖的公理系统的极限。

Carlson will work on the following four problems and related issues: (1) Do the strongly meager sets form an ideal? (2) How far along the inner model hierarchy do strong covering lemmas hold? (3) Is Fraisse's conjecture on models with many isomorphic substructures consistently false? (4) Can the existence of medial limits be proved in set theory? These are technical questions lying at the foundations of axiomatic set theory. One wants to know under what assumptions they are true, whether there are models of set theory in which they fail and others in which they hold. It is subtle questions such as these that allow us to explore the limits of the axiomatic systems upon which all of mathematics logically rests.