Mathematical Sciences: Set Theory and Analysis

卡尔森将研究以下四个问题及相关问题：(1)强弱集是否构成理想？(2)强覆盖引理在内部模型层次结构中有多远？(3) Fraisse关于具有许多同构子结构的模型的猜想是否始终是错误的？(4)在集合论中能否证明中间极限的存在性？这些都是公理集合论基础上的技术问题。人们想知道在什么样的假设下它们是正确的，是否有一些集合论的模型是错误的而另一些是正确的。正是这些微妙的问题，使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.