In the large scale geometry there are several definition of dimension introduced by Gromov. Perhaps the main one is the notion of asymptotic dimension which proved to be useful to study the Novikov Conjecture. The asymptotic dimension is especially good for study discrete metric spaces such as finitely generated groups. In the 90's to deal with large manifolds Gromov introduced the notion of macroscopic dimension, a large scale dimensional invariant which ignores discrete objects. He did it to restate his old conjecture on manifolds with positive scalar curvature previously formulated in terms of filling radii. Gromov's conjecture says that the universal cover of closed n-manifold which admits a Riemannian metric of positive scalar curvature has the macroscopic dimension less or equal to n-2. This project suggests a development of the macroscopic dimension theory and its comparison with the asymptotic dimension. The ultimate goal is to prove Gromov's conjecture.
The notion of dimension plays an important role in almost all areas of mathematics and sciences. When one studies an unknown object like our universe, or the configuration space of robot's motion, or the set of solution of some equation, etc., one of the first parameters he/she would like to know is dimension of the object. The intuitive idea of dimension has been adjusted rigorously to many areas of research. The project does dimension theory of the large scale world. If completed it will make a breakthrough on some old famous problems in Geometry and Topology.
