Many problems in "higher dimensional" topology have been reformulated in terms of one or more of the following theories: homotopy theory; algebraic K-theory; surgery L-theory; stable pseudoisotopy theory. In an effort to better understand the latter three theories, Jones (in collaboration with F.T. Farrell) has formulated an "Isomorphism Conjecture" for each of these three theories. For example, the Isomorphism Conjecture for algebraic K-theory gives a simple recipe for computing the algebraic K-groups for the integral group ring of a group G in terms of the algebraic K-groups for the integral group rings of all the infracyclic subgroups of G. (A group is "infracyclic" if it is a finite extension of a cyclic group.) Quite a bit is know about the algebraic K-groups for integral group rings of infracyclic groups; so the truth of the preceeding conjecture would help considerably in understanding the algebraic K-theory of all integral group rings. As part of this proposal, Jones intends to continue working towards verifying the Isomorphism Conjecture for algebraic K-theory of the integral group ring of G, under the condition that the group G acts properly, discontinuously by isometries on a complete, Riemannian manifold which has non-positive sectional curvature values everywhere.
Mathematicians study the structure of spaces by trying to find the "genetic code" for each space. The "genetic code" for a space usually comes in the form of an algebraic gadget associated to the space; typically there is a simple recipe for constructing the algebraic gadget from the space. The hope is that two spaces having equal or congruent "genetic codes" should be equal or congruent as spaces. One of the first algebraic gadgets to be associated to a space is called the "fundamental group" of the space. The French mathematician Henri Poincare introduced this idea about a 100 years ago. The fundamental group of a space is certainly an important ingredient in its "genetic code", but there are examples of many different spaces which have the same fundamental group; in these cases the fundamental group is not a complete "genetic code" for the space. About 50 years ago the mathematician Armand Borel (of the Institute for Advanced Study in Princeton, N.J.) conjectured that for an important class of spaces (called "aspherical spaces") the fundamental group does give the complete "genetic code" for the space; that is, two aspherical spaces with the same fundamental group are forced to be the same space. Jones has made some progress towards proving Borel's Conjecture, and will continue his work in that direction.
