**Alessandro Rapa, Università di** **Verona**

**Title****: ****Simple
objects in the heart of a t-structure**

**Abstract:** In this talk, we consider a
specific class of finite dimensional algebras of infinite
representation type, called "tubular algebras". Pure-injective
modules over tubular algebras have been partially classified
by Angeleri Hügel and Kussin, in 2016, and we want to give
a contribution to the classification of the ones of "irrational
slope". First, we move to a more geometrical framework,
i.e. we work in the category of quasi-coherent sheaves over a
tubular curve, and we approach our classification problem from
the point of view of tilting/cotilting theory. More precisely,
we consider the
Happel-Reiten-Smalø heart of torsion pairs cogenerated by infinite
dimensional cotilting sheaves. These hearts are locally coherent
Grothendieck categories in which the pure-injective sheaves over
the tubular curve become injective objects. In order to study
injective objects in a Grothendieck category is fundamental to
know the classification of the simple objects. We will use
techniques coming from the continued fractions and universal
extensions to provide a method to construct an infinite
dimensional sheaf of a prescribed irrational slope that becomes
simple in the Grothendieck category given as the heart of a
precise t-structure.