The interplay between algebra and logic

在这个项目中，我们专注于代数和逻辑之间的相互作用。在过去的几十年中，代数学家使用无限组合学和集合论以及模型论的方法来解决代数中长期存在的问题，这些方法分别与Zermelo-Fraenkel公理和选择公理（ZFC）给出的通常的集合论相一致。模型的构建具有某些组合原则，如预测原则，单值化原则等，通过强迫某些大基数存在的模型，发挥了重要作用。很多时候，代数性质等价于集合论的陈述，例如基数条件，如（广义）连续统假设的（非）有效性。这是这个项目的目的是进一步研究代数和逻辑之间的相互作用。特别是，我们将专注于无限秩巴特勒模块的结构，大基数，自同构塔问题和对偶群在各种模型的ZFC得到强迫。

In this project we focus on the interplay between algebra and logic. Over the last decades algebraists have used infinite combinatorics and set-theoretic as well as model-theoretic methods to solve long-standing problems in algebra by showing their consistency with, respectively independence of the usual set theory given by the Zermelo-Fraenkel axioms and the axiom of choice (ZFC). The construction of models with certain combinatorial principles like prediction principles, uniformization principles, etc. by forcing over models in which certain large cardinals exist, played an essential role. Very often algebraic properties are equivalent to set-theoretic statements, for instance cardinal conditions like the (non-)validity of the (generalized) continuum hypothesis. It is the aim of this project to further investigate this interplay between algebra and logic. In particular we shall focus on the structure of infinite rank Butler modules, large cardinals, the automorphism tower problem and dual groups in various models of ZFC obtained by forcing.