Machiel van Frankenhuijsen Knihy

Exploring the connections between number theory, spectral geometry, and fractal geometry, this study delves into the vibrations of fractal strings—one-dimensional drums characterized by fractal boundaries. The work presents a rigorous examination of how these mathematical concepts interrelate, providing insights into the properties and behaviors of fractal structures in the context of vibrations.

This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.