Page 66 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 66
ABEL LECTURE
Graph limits and Markov spaces
László Lovász, laszlo.lovasz@ttk.elte.hu
Eötvös Loránd University, Hungary
Limit objects for sequences of finite structures, larger and largerin size but more and more sim-
ilar in some sense, have been constructedsporadically, perhaps since von Neumann constructed
continuous geometries, butthis research has become quite extensive in the last decade and a
half. Graphsare perhaps the simplest structures, and accordingly, the limit theory ofgraphs has
made the most progress. The theory of graph limits is only understood, to a somewhat satisfac-
torydegree, in the cases of bounded degree graphs (initiated by Benjamini andSchramm) and
of dense graphs (initiated by Borgs, Chayes, Lovász, Szegedy, Sósand Vesztergombi). More
recently there is a lot of interest in the intermediatecases. It appears that the most important
constituents of graph limits in the general case will be Markov spaces (Markov chains on mea-
surable spaces with astationary distribution). Several important theorems can be extended from
finite graphs to Markov spacesor, more generally, to measurable spaces: flow theory, expanders
and spectra,mixing of random walks, etc. In this talk we will give a glimpse into thisemerging
theory, based on the work of Á. Backhaus, D. Kunszenti-Kovács, B.Szegedy and the speaker.
64
Graph limits and Markov spaces
László Lovász, laszlo.lovasz@ttk.elte.hu
Eötvös Loránd University, Hungary
Limit objects for sequences of finite structures, larger and largerin size but more and more sim-
ilar in some sense, have been constructedsporadically, perhaps since von Neumann constructed
continuous geometries, butthis research has become quite extensive in the last decade and a
half. Graphsare perhaps the simplest structures, and accordingly, the limit theory ofgraphs has
made the most progress. The theory of graph limits is only understood, to a somewhat satisfac-
torydegree, in the cases of bounded degree graphs (initiated by Benjamini andSchramm) and
of dense graphs (initiated by Borgs, Chayes, Lovász, Szegedy, Sósand Vesztergombi). More
recently there is a lot of interest in the intermediatecases. It appears that the most important
constituents of graph limits in the general case will be Markov spaces (Markov chains on mea-
surable spaces with astationary distribution). Several important theorems can be extended from
finite graphs to Markov spacesor, more generally, to measurable spaces: flow theory, expanders
and spectra,mixing of random walks, etc. In this talk we will give a glimpse into thisemerging
theory, based on the work of Á. Backhaus, D. Kunszenti-Kovács, B.Szegedy and the speaker.
64
