Around 2004, Lovasz and Szegedy came up with a certain compactification of the space of finite graphs.
More precisely, they proved that there exists a metric - now called the cut-distance - which yields a
compact topology. Their proof of compactnes relies on the Szemeredi regularity lemma. An entire theory,
with applications in extremal graph theory and random graphs, developed from this statement. I will
talk about approaching the cut-norm topology via the weak* topology. This approach gives a new view of
many properties, and in particular yields a quick and elementary proof of the Lovasz-Szegedy theorem.
The talk will be self-contained. Various bits of this are joint work with Martin Dolezal, Jan Grebik,
Jon Noel, Diana Piguet, Israel Rocha, Vaclav Rozhon, Maria Saumell.