Noncommutative geometry and number theory

Noncommutative geometry is a modern field of mathematics begun by Alain Connes in the early 1980s. It provides powerful tools to treat spaces that are essentially of a quantum nature. Unlike the case of ordinary spaces, their algebraof coordinates is noncommutative, reflecting phenomena like the Heisenberg uncertainty principle in quantum mechanics. What is especially interesting is the fact that such quantum spaces are abundant in mathematics. One obtains them easily when one considers equivalence relations that are so drastic that they tend to collapse most points together, yet one wishes to retain enough information in the process to be able to do interesting geometry on the resulting space. In such cases, noncommutative geometry shows that there is a quantum cloud surrounding the classical space, which retains all the essential geometric information, even when the underlying classical space becomes extremely degenerate. It is to this quantum aura that all sophisticated tools of geometry and mathematical analysis, properly reinterpreted, can still be applied. It has become increasingly evident in re cent years that the tools of noncommutative geometry may find new and important applications in number theory, a very different branch of pure mathematics with an ancient and illustrious history. This has happened mostly through a new approach of Connes to the Riemann hypothesis (at present the most famous unsolved problem in mathematics).