Some connections between discrete and continuous mathematics, Mirna Dzamonja

Part of the mathematics pure colloquium.

Recent years have seen the developments by Lovasz and others of graphons, in which the theory of finite graphs has met the theory of measurable functions from [0,1]^2 to [0,1]. Techniques developed in this area allow a transfer of results about graphs to results about graphons and vice versa. A posteriori, by the results of Elek and Szegedy, one can see that in fact graphons are an instance of constructing an ultraproduct measure on an ultraproduct of finite structures, so endowed with a counting measure. A much more general context including this one was actually studied by Loeb in the 1970s, who constructed ultraproduct measures on general measure spaces. We are particularly interested in the context of the ultraproduct of Boolean algebras, which allow us to give a graphon-like version of Boolean algebras. We connect this with our work from 2014 where we modelled the universality spectrum of Banach spaces under isomorphisms using an ultraproduct of Boolean algebras.

Mirna Dzamonja, UEA