Abstract: We give an introduction to the Standard Model of particle physics, as it is derived in the mathematical framework of noncommutative geometry. Using not much more mathematics but matrix multiplication, the emergent picture of the fine structure of spacetime is that of the product of the
continuum with a noncommutative finite space. The latter describes the internal degrees of freedom, which are now translated in terms of geometry.
In the second part of the talk, we will discuss some of the physical consequences of our approach. We argue that there is a natural candidate to go Beyond the Standard Model, involving a Pati-Salam gauge symmetry at higher energy. We justify our approach by analyzing the running of the Pati-Salam
gauge couplings, with boundary conditions set by the (usual) Standard Model couplings at an intermediate mass scale at which the Pati-Salam symmetry is broken. We find that the three Pati-Salam gauge couplings meet in a single point and hence establish grand unification.
We conclude by presenting some recent results that links the spectral action to entropy, which could indicate a path to a more intrinsic, noncommutative geometric formulation of the quantum theory.