Final Report Summary - GETOM (Geometry and Topology of Open Manifolds)
The starting point of the grant is a question asked, in 1904, by the great french mathematician Henri Poincaré. The question has been answered in 2003 by the russian mathematician Grigori Perelman. It concerns the structure of 3D objects which are closed, that is bounded in space, and without boundary and which satisfy a property called simply connectedness. This fancy words mean that every closed curve drawn on the space can be continuously deformed to a point, without breaking it. The so-called Poincaré conjecture asserted that the only 3D space with these properties is the 3D-sphere. This assertion turned out to be true and was proved by Grigori Perelman in 2003. Perelman got the Fields medal in 2010 (which he refused). The case of open spaces, that is 3D-spaces which are infinite, was not touch at all and the purpose of the grant was to start the study of such spaces. Particurlarly those which are called contractible, that is those which can be contracted continuously (whithout breaking) to a point. It is quite clear that the standard Euclidean space is contractibe but it turns out that there are lots of such spaces, indeed uncountably many, whose geometry is widely unknown. The purpose of the grant was to study them. One of the outcome is that such spaces do not admit any Riemannian metric with positive scalar curvature unless it is the Euclidean space. It is a beautiful rigidity result, that is a result saying that only one space has the desired property, here positive scalar curvature. This is proved by Jian Wang a graduate student supported by the grant.