For a \(\sigma\)-ideal \(I\) on the Baire Space, a real is called quasi-generic over a model of set theory \(M\) if it avoids all Borel \(I\)-small sets coded in \(M\). This is a natural generalisation of Cohen and Random reals due to Solovay, but only appeared as an explicit concept in the work of Brendle, Halbeisen and Löwe in 2005.
With the help of quasi-generic reals we can uncover connections between different phenomena in set theory of the reals, such as cardinal invariants, regularity properties for sets in the projective hierarchy, and forcing-theoretic properties of partial orders.
In this talk, I will first survey older results regarding this connection, built up using the uniform framework of idealized forcing due to Jindřich Zapletal. Then, I will survey recent developments in this area and mention several interesting problems that are still open.