The purpose of this book is to explain the relevance of the subject of non-commutative geometry in dealing with two problems - the first, motivated by physics, is a subject of quantum gravity and the second, coming from mathematics, is the Riemann hypothesis. It is perhaps more aimed at mathematicians than physicists and the material is presented as much as possible in a self-contained way. The book consists of four chapters. The first chapter reviews all the standard aspects of quantum field theories, culminating in the construction of the standard model via non-commutative geometry and the geometric structure hidden behind renormalization. The second chapter describes a spectral realization of the zeros of the Riemann zeta function via non-commutative geometry. The third chapter introduces a realization of the geometry of adèle class space inside a quantum mechanical statistical system as an example of the non-commutative adèlic quotient. Based on the interplay of geometry, thermodynamics and the cohomology of motives, the last chapter clarifies in more detail the previously mentioned spectral realization of zeros of L-functions.

Reviewer:

pso