Noncommutative Geometry, the Spectral Standpoint

Abstract

We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions.