Noncommutative Geometry at the Newton Institute

This award is for partial support of US participants in a semester long workshop at the Newton Institute, Cambridge, UK, in noncommutative geometry for the period July 24 through December 22, 2006. Noncommutative geometry aims to carry over geometrical concepts to a new class of spaces whose algebras of functions are no longer commutative. The central idea goes back to quantum mechanics, where classical observables such as position and momenta no longer commute. In recent years it has become appreciated that such noncommutative spaces retain a rich topology and geometry expressed first of all in K-theory and K-homology, as well as in finer aspects of the theory. The subject has also been approached from a more algebraic side with the advent of quantum groups and their quantum homogeneous spaces. The subject in its modern form has also been connected with developments in several different fields of both pure mathematics and mathematical physics. In mathematics these include fruitful interactions with analysis, number theory, category theory and representation theory. In mathematical physics, developments include the quantum Hall effect, applications to the standard model in particle physics and to renormalization in quantum field theory, models of spacetimes with noncommuting coordinates. Noncommutative geometry also appears naturally in string/M-theory. The program will be devoted to bringing together these different streams and instances of noncommutative geometry, as well as identifying new emerging directions. Three main themes of the program will be reflected in workshops in July, September and December of 2006, covering noncommutative geometry and cyclic cohomology, noncommutative geometry and fundamental physics, and new directions in noncommutative geometry respectively.