Contributed talk: Refined $TC^-$ over $ku$ and derived q-Hodge complexes

As a consequence of his proof of rigidity of the category of localising motives, Efimov has constructed refinements of localising invariants. Such refined invariants often contain a lot more information than the original ones. For example, the refined $TC^-$ of the rational numbers is not a rational spectrum; it contains very subtle p-complete information as well. In this talk, we'll explain how to compute it after base change to $ku$. The computation involves a surprising connection to q-de Rham cohomology, and in partcular, to the question of whether there exists a "q-Hodge filtration" on q-de Rham cohomology. This is joint work with Samuel Meyer.