3-Manifolds and number fields

DMS-0307078 Adam Sikora Thirty years ago B. Mazur discovered some surprising similarities (a) between knots and prime numbers, and (b) between 3-dimensional manifolds and number fields. Those similarities were further elaborated by Kapranov, Morishita, Ramachandran, Reznikov, and PI, but the full scope of these analogies is still unknown. PI's goal is to continue investigation of these analogies and their roots. Arithmetic topology is an exiting new area of mathematical research relating two of the most active areas of mathematical research in the recent years: low-dimensional topology and number theory. The development of Arithmetic Topology is driven by the desire to establish rigorous and uniform foundations for those two seemingly diverse theories.