## Seminar (Operator Algebras)

Joan Bosa (Universitat Autònoma de Barcelona) delivered the talk:

Villadsen Algebras: Projections and Vector Bundles

Abstract:

Les àlgebres de Villadsen són un tipus de C*-àlgebres que van ser utilitzades per trobar contraexemples a la conjectura de Classificació d’Elliott. Per provar que la conjectura fallava van utilitzar que l’ordre de les projeccions sobre espais topologics s’associa a l’ordre entre els vector bundles d’aquests. Així, en les àlgebres de Villadsen s’utilitza fortament la teoria de vector bundles per tal de construir l’exemple dessitjat. En aquesta xerrada explicarem una mica la història de la classificació de C*-àlgebres, i donarem algunes pinzellades sobre com utilitzar la teoria de vector bundles i les classes de Chern al món de les C*-àlgebres.

## Seminar (Ring Theory)

Giovanna Le Gros (Università di Padova) delivered the talk:

Minimal approximations and 1-tilting cotorsion pairs over commutative rings

Abstract:

Minimal approximations of modules, or covers and envelopes of modules, were introduced as a tool to approximate modules by classes of modules which are more manageable. For a class C of R-modules, the aim is to characterise the rings over which every module has a C-cover or C-envelope. Moreover A-precovers and B-preenvelopes are strongly related to the notion of a cotorsion pair (A,B).

In this talk we are interested in the particular case that (P_1,B) is the cotorsion pair generated by the modules of projective dimension at most one (denoted P_1) over commutative rings. More precisely, we investigate over which rings these cotorsion pairs admit covers or envelopes. Furthermore, we interested in Enochs’ Conjecture in this setting, that is if P_1 is covering necessarily implies that it is closed under direct limits. The investigation of the cotorsion pair (P_1,B) splits into two cases: when the cotorsion pair is of finite type and when it is not. In this talk I will outline some results for the case that the cotorsion pair is of finite type, where we consider more generally a 1-tilting cotorsion pair over a commutative ring.