**Bernhard Keller**

Université Paris 7

Cluster algebras were invented by S. Fomin and A. Zelevinsky in
spring 2000 as a tool to approach Lusztig's theory of canonical
bases in quantum groups and total positivity in algebraic groups.
Since then, cluster algebras have become the center of a rapidly
developing theory, which has turned out to be closely related to a
large spectrum of other subjects, notably Lie theory, Poisson
geometry, Teichmüller theory and quiver representations. Recent
work by many authors has shown that this last link is best
understood using the cluster category, which is a triangulated
category associated with every Dynkin diagram. In this talk, I will
report on these developments and present the cluster
multiplication theorem, obtained in joint work with P. Caldero,
which directly links the multiplication of the cluster algebra to
the triangles in the cluster category.

2005-05-23