Habiro Cohomology

I got interested in q-deformations when I realized that de Rham cohomology seems to admit a canonical q-deformation: p-adically, this follows from prismatic cohomology. It was a natural question whether this q-deformation -- a priori only defined after completion at q=1 -- is defined after completing at all roots of unity, i.e. over the Habiro ring. This led in particular to the definition of the Habiro ring of number fields, of which Garoufalidis and Zagier found explicit elements: This was the starting point of our joint work, and a lot of inspiring discussions. In this talk, I want to explain a larger theory of "Habiro cohomology" (very roughly, a q-deformation of de Rham cohomology to the Habiro ring); this includes notably the work of Ferdinand Wagner.