Categorical noncommutative geometry and localizing motives

I will give a brief overview of some notions and results on localizing invariants of stable infinity-categories (over some base ring). Mostly I will focus on my recent results about the category of localizing motives -- the target of the universal localizing invariant commuting with filtered colimits. Most strikingly, this category turns out to be rigid as a "large" symmetric monoidal category, in the sense of Gaitsgory and Rozenblyum. An important role in the proof is played by the categorical notions of smoothness and properness, and by the existence of categorical smooth compactifications in certain situations. If time permits, I will discuss the applications of rigidity of localizing motives, such as refined versions of (topological) Hochschild and negative cyclic homology and their uses.