Balázs Szegedy: Action convergence of operators and graphs

In the past two decades a new field called "graph limit theory" started to emerge. This rather rich subject is at the meeting point of many different fields including graph theory, analysis, probability theory and algebra. In a nutshell, graph limit theory is the study of the completion of the set of finite graphs in a given similarity metric. It brings a powerful analytic view point into discrete mathematics. Depending on applications, various non-equivalent metrics have been used and they led to non-equivalent limit concepts such as dense graph limits, Benjamini-Schramm limits, local-global limits, Lp convergence, shape convergence, logarithmic limits etc... In this talk we give a new approach to the subject which unifies many of these limits based on a limit theory for operators acting on random variables. As a first application we show a new approach to random matrix theory. Joint work with Agnes Backhausz.