Principal Investigator: Jean-Francois R. Lafont
The Principal Investigator (PI) proposes to work on a series of projects that incorporate techniques from geometry and topology, with a view to applications to both of these fields. A major component of this proposal focuses on making use of geometric arguments to shed light on the algebraic K-theory of (integral group rings of ) various classes of groups. The PI's previous work with Ortiz provided a complete algorithm to determind the lower algebraic K-theory for lattices in the isometry group of hyperbolic 3-space. The PI intends to further develop techniques for computing the lower algebraic K-theory of various other geometrically significant classes of groups. The importance of such explicit computations is twofold. First of all, it provides a testing ground for conjectures, and a possible source of new conjectures. Secondly, because of the close relationship between algebraic K-theory and high-dimensional topology, we expect our computations to shed light on the topology of high-dimensional manifolds. The PI is also interested in better understanding how various metric properties constrain the topology of the underlying space. There is a lot of freedom here, depending on what one means by a metric property, and what aspect of the topology we choose to study. For instance, one can consider how Riemannian curvature influences the bounded cohomology of a space. Or one can try to study how metric properties (e.g. the G-space axioms) can influence the local topology of a space. Alternatively, one can try to study, within a fixed class of topological spaces (e.g. closed smooth manifolds), those supporting various types of metric properties (e.g. Riemannian metrics of non-positive curvature, as compared to locally CAT(0) metrics). The PI intends to work on questions related to this circle of ideas.
Geometry is the study of properties of spaces equipped with some additional structure, usually of a metric nature (i.e. allowing one to measure distances in the space). In the field of topology, one forgets about the precise underlying distance, and only retain the notion of points in the space being "close together". As such, topology is often referred to as "rubber geometry": one can stretch and deform spaces without changing their underlying topology. In topology, manifolds are the spaces of most interest. These are spaces which, on the small scale, look like Euclidean space, but on the large scale, can be curved. A basic question lies in finding ways to detect when two manifolds are different from each other. This is usually done by associating certain invariants to manifolds: if the invariants are different, then the manifolds are different. One family of invariants which is particularly useful in high-dimensional topology are the algebraic K-theory invariants. In general, these invariants are quite hard to compute. The PI plans to develop methods for computing these invariants, in the special case where the manifold has some nice underlying geometry. More generally, the PI plans on studying how the presence of a geometric structure constrains various topological properties (and vice versa).
