Algebraický seminář
Chris Lambie-Hanson (IM CAS, Prague): Whitehead's problem and condensed mathematics
Abstract: Whitehead's problem, which asks whether every Whitehead abelian group is a free abelian group, was a prominent open question in group theory in the mid-20th century. In 1974, Shelah proved that the problem is independent of ZFC, which was a surprising development and provided one of the first instances of a major problem coming from outside logic and set theory to be proven independent of ZFC. In recent years, Clausen and Scholze have introduced the framework of condensed mathematics as a setting in which to apply algebraic tools to the study of algebraic structures carrying topologies. Through some deep structural analysis of the category of condensed abelian groups, they showed that, when appropriately interpreted, Whitehead's problem is not independent in this setting: it is provable in ZFC that every abelian group that is Whitehead in the condensed category must be free. In this talk, we sketch a new, more concrete proof of Clausen and Scholze's result, in the process highlighting some connections between condensed mathematics and the theory of set-theoretic forcing. We will end by considering some variations on this question in the context of condensed or solid abelian groups. This is joint work with Jeffrey Bergfalk and Jan Šaroch.
The Algebra Seminar was founded by Vladimir Korinek in the early 1950's and continued by Karel Drbohlav until 1981. The seminar resumed its activities in 1990 under the guidance of Jaroslav Jezek and Tomas Kepka. Since 1994, the seminar is headed by Jan Trlifaj.
Presently, the seminar is supported by GACR. It serves primarily as a platform for presentation of recent research of the visitors to the Department of Algebra as well as members of the Department and their students.
