Mandi Schaeffer Fry (Metropolitan State University of Denver)
X
Widely known as the founder of modular representation theory, Richard Brauer set the stage for the study of the so-called local-global conjectures in the character theory of finite groups. These are conjectures that relate the representations and structure of a finite group to that of certain proper (“local”) subgroups. In this talk, I'll discuss some of these conjectures and their implications for other character-theoretic properties and lingering questions of Brauer.
I will sketch a proof that, assuming 0† does not exist, if there is a partition of R into ℵω Borel sets, then there is also a partition of R into ℵω+1 Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of ℵω.) This contrasts starkly with the situation for cardinals with uncountable cofinality and their successors, where the spectrum of possible sizes of partitions of R into Borel sets can be made completely arbitrary via forcing.