### Deformation Theory Seminar

Monday, October 18, 2021 - 2:00pm

#### Alfredo Najera

UNAM- Oaxaca

Location

University of Pennsylvania

zoom

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).