## Logic Seminar

Recently, Clausen and Scholze introduced the framework of condensed mathematics as a convenient setting in which to apply algebraic tools to the study of algebraic objects carrying topologies. The basic idea is to embed (large portions of) classical categories, such as the category of topological abelian groups or of topological vector spaces, into richer categories, consisting of condensed versions of the classical objects, that enjoy nicer algebraic properties. While developing the theory of condensed abelian groups, Clausen and Scholze proved that, when appropriately interpreted, Whitehead's problem is no longer independent when asked in the context of condensed abelian groups. Namely, if an abelian group is Whitehead in the category of condensed abelian groups, then it must be free. In this talk, we will give a brief introduction to condensed mathematics, present an outline of a new (though very much building on Shelah's seminal work in the classical setting), combinatorial set theoretic proof of Clausen and Scholze's result, and discuss some more general connections between condensed mathematics and forcing that are brought to light by the proof. This is joint work with Jeffrey Bergfalk and Jan Šaroch.