Atiyah and Singer proved their famous "index theorem" in topology using ideas and methods directly inspired by Grothendieck's proof of his huge generalization of Riemann-Roch's theorem, a proof that makes a prototypical use of several of the newest aspects of modern (that is Grothendieck's) algebraic geometry, for example the insistence in working in a relative rather than absolute situation, that is for morphisms rather than simply for objects.