In recent years ideas from geometry have driven some of the most exciting developments in combinatorics such as Gromov hyperbolic groups and CAT(0) spaces, combinatorial Morse theory, combinatorial Ricci curvature, combinatorial differential manifolds and matroid bundles. The central unifying notion in geometry is that of curvature. Now, through these diverse geometric and combinatorial theories, curvature is emerging as a powerful tool and fundamental unifying concept in combinatorics as well. This Focused Research Group will explore some of the specific notions of combinatorial curvature driving current combinatorial work, and also the role of curvature as the basis for a coherent geometric vision of combinatorics itself.
The notion of curvature has been one of the grand unifying concepts in geometry and physics for well over a century. For example, Gauss, the originator of our modern understanding of curvature, showed that Euclidean geometry was distinguished from other geometries as being the geometry of a space with zero curvature. As an application he showed that it is precisely the curvature of the surface of the Earth which makes it impossible to draw a map of the Earth's surface (on a flat piece of paper) that accurately portrays all lengths and angles. Riemann generalized Gauss's work to smooth spaces of higher dimensions, and Einstein observed that Riemannian geometry was precisely the right setting in which to describe his theory of general relativity (in which the curvature of the universe is the result of gravitational forces). Partly as a result of Einstein's work, the last century saw an intensive investigation into the curvature of smooth spaces. Combinatorics, roughly defined, is the study of objects which can be described by a finite amount of information. This is precisely the mathematics that computers can do. This type of mathematics seems far removed from the geometric investigations of Gauss, Riemann and countless others. However, there is a growing collection of combinatorial phenomena which can best be viewed as being finite analogues of facts about the curvature of smooth spaces. The goal of this proposal is to come to a coherent understanding of curvature as a combinatorial notion. In addition, bringing together researchers from a variety of mathematical disciplines, we wish to bridge the chasms between geometry, combinatorics, algebra and topology, using curvature as the unifying theme.
