Fields institute,Room 210
Speaker: David J. Fernández Bretón
Title: “Every strongly summable ultrafilter is sparse!”
Abstract: The concept of a Strongly Summable Ultrafilter was born from Hindman’s efforts for proving the theorem that now bears his name (which at the time was known as Graham-Rothschild’s conjecture), although later on it got a life of its own and started to be studied for its own sake, mostly because of its nice algebraic properties. At the time the focus was on ultrafilters over the semigroup $(\mathbb N,+)$, but eventually Hindman, Protasov and Strauss generalized much of this theory to abelian groups in general in a 1998 paper. In that same paper, they introduced the notion of a sparse ultrafilter, one which subsumes that of strongly summable as a particular case but that has even nicer algebraic properties. In a 2012 paper, Hindman, Steprans and Strauss found a large class of abelian groups (which included $(\mathbb N,+)$) over which every strongly summable ultrafilter must be sparse.
In this talk I extend this result to all abelian groups. Moreover we show that in most cases the strong summability of these ultrafilters is due to their being additively isomorphic to a union ultrafilter (I will explain what this means). However, this does not happen in all cases: I will also construct (assuming $\mathfrak p=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.