  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.
