Mathematical Sciences: Topology

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This project involves three senior researchers working in two areas of algebraic and geometric topology: (1) Automorphism groups of free groups, from the viewpoint of homological algebra and geometric group theory. Here the investigators have recently begun the development of a new three-dimensional model for automorphisms of free groups, refining the standard model, which is essentially one-dimensional. (2) Controlled and equivariant topology of manifolds and generalized manifolds, in particular, the study of the properties of the new class of higher-dimensional generalized manifolds recently discovered by Bryant, Ferry, Mio, and Weinberger. Group theory, the mathematical study of symmetries, is a pervasive thread in almost all the hard sciences, from quantum physics to data encryption. Free groups are the least constrained types of groups, from which all other groups are obtained by adding special relations. Hence the study of the symmetries, or automorphisms, of free groups has a particular significance for group theory. Manifolds are the global mathematical models of the universe we live in. Recent developments in theoretical physics, such as string theory, show that one should not confine one's attention just to three dimensions or even the four dimensions of space-time; more dimensions often provide a more accurate model, even if we cannot see them with our eyes.