Geometric and Algebraic Topology

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This project examines various aspects of the topology of manifolds: their classification, their bundles, and their symmetries. The project will focus on six areas. The first is to give a systematic approach to topological equivariant rigidity, using tools from surgery theory, algebraic K-and L-theory, and the Farrell-Jones Conjecture. The second is to study characteristics classes of matroid bundles (defined by Anderson and Davis) and to apply them to combinatorial incidence geometry. The third area is to investigate a rigidity conjecture involving self-homotopy equivalences of 3-manifolds and its connections with high-dimensional topology. The fourth area is to give the the classification, up to homeomorphism, of manifolds having the homotopy type of the total space of certain torus bundles over lens spaces. This is an application of the Farrell-Jones Conjecture. The fifth area is to compute the L-groups of a free product of groups, and thereby solve the connected sum problem - when is a manifold which is homotopy equivalent to a connected sum itself a connected sum. The last area is to study the algebraic and point-set topology of actions of p-groups on the torus from the point of view of homotopical group actions and Smith theory.Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Manifold theory connects with most areas of mathematics, as well as with physical phenomena such as cosmology, string theory, and classical and quantum mechanics. To understand and classify manifold one uses a variety of tools including algebraic topology, bundle theory, and differential geometry.