In this talk, dedicated to Paul Baum on the occasion of his 90th birthday, I will retrace a mathematical journey that began with our first encounter at the 1980 Kingston conference. The first part of the lecture will revisit the genesis of the Baum-Connes assembly map. We will explore how the interplay between homotopy theory and noncommutative geometry yields profound consequences that split naturally into geometry (via the injectivity of the map) and analysis (via its surjectivity), extending to the representation theory of Lie and $p$-adic groups.

In the second part of the talk, we will venture "beyond." I will explain why the topos-theoretic viewpoint is strictly more refined than the homotopy viewpoint. Then we shall deal with the key example of the "scaling site," discovered in my joint work with Katia Consani. It is the topos defined as the semidirect product of the half real line by the multiplicative action of the positive integers. I will illustrate how this topos serves as the mathematical idealization of a musical instrument, where the half-line encodes frequencies and the action of integers encode harmonics. We will see that the points of this scaling site coincide exactly with the noncommutative space which is the Riemann sector of the adele class space. This identity establishes one of the deepest relations I know between mathematics and music, giving direct access to the zeros of the Riemann zeta function as an absorption spectrum—without any need to define the zeta function itself.