Principal Investigator: Igor Mineyev
The main objective of this project is to develop techniques for constructing metric spaces. One would like to have nice metric spaces on which a given group can act, either by isometries or topologically; and to make them "as canonical as possible". The principal investigator's symmetric join construction and the conformal metric on the boundaries of hyperbolic groups, and others, were steps in this general direction. Ideally, one would like to have a functorial construction that associates to every group in a given class (with a metric on it), say, a uniquely geodesic metric space with a (cocompact, isometric, properly discontinuous) action by the group. This seems to be a very difficult question (this is the kind of things that Perelman's Ricci flow does to manifolds, and we essentially want its analogue for general metric spaces). This question is related to several major problems in topology and geometry. Our goal is to make further progress in this direction. Another goal is to further investigate how geometry of groups is reflected in various types of (co)homology, and vice versa.
A metric space is one of the most important and most basic mathematical notions: it is a set of points in which one is given the distance between any two points, with very natural properties. All branches of mathematics use this notion. Our Universe is an example of a metric space, and there is no reason for it to look like the 3-dimensional Euclidean space at all. We are interested in understanding what kinds of metric spaces can possibly be there, and "how good" they can be. For example, in order to be "good", one should at least be able to pass from one point to another by a shortest possible path within the given space. Manifolds considered by classical Riemannian geometry have this property. We aim for a generalization of classical geometry from manifolds to metric spaces. One goal of this project is to explicitly construct good metric spaces and to understand how they form. This simple idea is related to deep questions facing modern mathematics. As a by-product, this might give us more insight to the structure and formation of our Universe. Our original motivation though is the beauty and mathematical importance of the problem rather than any particular immediate application. (Our Universe is only a special case of a metric space.)
