Cluster Theory

It was initiated by Fomin und Zelevinsky when they introduced "cluster algebras"
- these are certain integral domains with a predescribed set of finite subsets (the clusters).

The cluster theory relates to

and other parts of mathematics!

The corresponding cluster complex is based on the set of almost positive roots
(for a Kac-Moody Lie-algebra, taking into account an ordering of the root basis.)

The smallest non-trivial cluster algebra was discussed already by Gauss under the name Pentagramma Mirificum.