Singularities in the Grothendieck ring of varieties and motivic zeta functions

For a singular variety $X$, one can define several “motivic” types of singularities using the Grothendieck ring of varieties. One of them is the notion of $\mathbb L$-rational singularities, introduced by Nicaise and Shinder in their work on the specialization of stable birationality. The question of when quotient varieties $X/G$ have $\mathbb L$-rational singularities turns out to be particularly interesting. I will discuss some results in this direction, in particular a proof that symmetric powers of smooth varieties have $\mathbb L$-rational singularities. I will then explain how one can use this result to prove the irrationality of motivic zeta functions for a large class of varieties, generalizing a method of Larsen and Lunts to higher dimensions. If time permits, I will also discuss some other applications and open questions.