Abstract: I will explain why the étale fundamental group of a smooth projective variety is finitely presented, following the recent work in arXiv:2102.13424.
Abstract: When X is a compact Hausdorff space and C(X) is the ring of continuous complex-valued functions on X, K_0(C(X)) coincides with topological K-theory of X, K^0(X). However, for n different from 0, K_n(C(X)) Is not the same as topological K-theory. The exact relationship between the two has been a subject of study for 40 years. I will explain a bit of the history and how this recent paper of Aoki largely settles the outstanding questions.