Dimension theory on large scales and its applications

University Of Florida, Gainesville FL

Investigators

Abstract

The large scales dimension theory is a part of coarse geometry, a relatively new subject which studies large scale properties of metric spaces. It is a growing discipline which has some analogy with the classical dimension theory which is a part of topology, a science of small scales. Many invariants of the large scale dimension theory such as the asymptotic dimension and the macroscopic dimension proved to be useful in studying finitely generated groups and open Riemannian manifolds. Thus, the large scale dimension appears in different areas of mathematics such as topology of manifolds, group theory, differential geometry. One of the main goals of this project is to prove the Gromov Conjecture about manifolds with positive scalar curvature. Another application of this project is a further development of a dimension theoretic approach to the Novikov Higher Signature Conjecture. The classical topology has a century long history and it is substantially developed in all directions. It became a fundamental core part of mathematics. Despite on abstractness it has many profound applications to virtually all sciences (Physics, Chemistry, Biology, Economics, etc.). The large scale version of topology and in particular the large scale analogue of dimension theory has a potential to grow in a discipline of similar weight and to bring numerous application beyond the scope of Mathematics. Thus the ideas of coarse geometry are already used in the large data mining. Clearly, a quantitative processing of overwhelming flaws of data of different kind will be more and more on demand in all activities of the mankind. Another area of application of dimension theory which is addressed in this project is studying the complexity of robot motion planning algorithms.