The interplay between algebra and logic

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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.