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
- Canonical bases, dual canonical bases
- Lusztig's positivity results
- Polytopes (Stasheff)
- Toric geometry
- Teichmüller theory
- Representation theory
- Calabi-Yau categories
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.