grothendieck’s standard conjectures, voevodsky’s nilpotence conjecture, and tate’s conjecture, play a key central role in algebraic geometry. notwithstanding the effort of several generations of mathematicians, the proofs of these celebrated conjectures remain ellusive. the aim of this talk, prepared for a broad audience, is to give an overview of a recent topological/noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.