U.S.-Japan Cooperative Research: Primes and Knots

Johns Hopkins University, Baltimore MD

Investigators

Abstract

0124616 Morava This award supports the participation of American scientists in a U.S.-Japan seminar on primes and knots to be held at the Johns Hopkins University in Baltimore, Maryland from March 15-22, 2003. The co-organizers are professors Jack Morava of the Johns Hopkins University and Professor Toshitake Kohno at the University of Tokyo in Japan. The theory of primes and the theory of knots are perhaps the most venerated branches of algebra and of topology, and in many ways, they are both still the most accessible. However, it is only relatively recently that researchers have begun to perceive deep relations between them, via analogies between the Galois groups of number fields and the fundamental groups of link complements. The topic has the advantage of being approachable from many directions and on many levels. The theory of knots and the theory of primes are both intuitively accessible, and the participants expect this seminar to foster the development of a common language between researchers in these areas. A prime resembles a knot, and the ideal generated by an algebraic integer is like the boundary of an embedded surface. In both subjects, relations between abelian constructions such as Alexander invariants are relatively well-understood, and much current research centers on deeper nonabelian questions; for example, it is now known that {5, 41, 61} is the first set of Borromean primes (in which no two are nontrivially linked, but all three are). Seminar organizers have made a special effort to involve younger researchers and graduate students as both participants and observers. The exchange of ideas and data with Japanese experts in this field will enable U.S. participants to advance their own work, and will set the stage for future collaborative projects. Dissemination of information on the seminar will be available on the World Wide Web.