Talk 1: Refined $TC^-$ over perfectoid rings

For p-adic formal schemes, there is a tight link between prismatic cohomology and $TC^-$. A little more than a year ago, I was trying to understand how prismatic cohomology should work for rigid-analytic varieties, and realized that Efimov's result that localizing motives form a dualizable category allows one to decomplete the functor $TC^-$. This refined $TC^-$ induces a nontrivial functor for localizing motives over the generic fibre; yielding in particular such a functor for rigid-analytic varieties over $\mathbb{C}_p$ (by applying it to the dualizable category of nuclear modules). I will explain this construction, and will try to say something about the (still somewhat mysterious) relation to the (now existent) theory of prismatic cohomology for rigid-analytic varieties.