The purpose of this talk is to explain a fruitful

interaction between ideas/constructions coming form the theory of

cluster algebras, representation theory of associative algebras and

deformation theory.

The representation theory of associative algebras is a well developed

branch of mathematics that has been very active for nearly 50 years.

The theory of cluster algebras is much younger, it was initiated by

Fomin and Zelevinsky in 2001. Various important developments in these

theories have emerged in the last 15 years thanks to the deep relation

that exists in between them. After an introduction to this circle of

ideas I will recall the construction of a simplicial complex $

\Delta(A)$ -- the \tau-tilting complex-- associated to a finite

dimensional associative algebra $A$. Then I will report on one aspect

of work-in-progress with Nathan Ilten and Hipólito Treffinger. We show

that if $ \Delta(A)$ is a cluster complex of finite type then the

associated cluster algebra with universal coefficients is equal to a

canonically identified subfamily of the semiuniversal family for the

Stanley-Reisner ring of $ \Delta(A)$. Time permitting I will elaborate

on some aspects of the "non-cluster" case (namely, when $\Delta(A)$

is not a cluster complex).