Classification of cosilting modules in type $\tilde{A}$

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.