Algebraic and geometric topology

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Our physical universe is modelled as a 3-dimensional geometric object called a 3-manifold, analogous to 2-D surfaces.  Unlike surfaces, classifying the possible types of 3-manifolds is still an open and challenging problem.Much of the information on 3-manifolds is encoded in algebra, for example the so-called fundamental group.  Despite spectacular recent progress, such as the solution of the Poincare conjecture, which identifies the only 3-manifold which has trivial fundamental group, many open questions and mysteries remain in 3-D topology regarding manifolds and objects in them such as knots and surfaces.This proposal is to study the deep connection between 3-D topology and the algebraic constructions which reflect their shapes and structures within them.  It is expected that this will find application in such diverse areas as cosmology, dynamical systems, and other sciences such as chemistry and biology in which understanding 3-D stuctures is important.