Combinatorial Differential Geometry and Topology

William Marsh Rice University, Houston TX

Investigators

Abstract

Robin Forman is continuing his work on the global geometry and topology of combinatorial spaces. The emphasis is on adapting techniques from the study of smooth manifolds. In particular, he has developed combinatorial versions of Morse theory and Bochner's method, two classical techniques from differential topology and differential geometry, respectively. Many others have approached the study of combinatorial spaces from this point of view, but it seems that Forman's approach is simpler than that which has come before, and also retains more of the flavor of the motivating subjects of differential topology and geometry. Numerous applications have been found to the study of combinatorial spaces arising in various aspects of algebra, graph theory, and complexity theory. The ultimate goal is a unification of the study of smooth and combinatorial spaces. Compact combinatorial spaces arise in a wide variety of settings in discrete mathematics, and can be described, in a very convenient way, with a finite amount of data describing the local structure near each point. There has been a resurgence of interest in such spaces because they are well suited for computer experimentations and calculations. Moreover, in numerous real world settings one has the goal of investigating a space, but the only available information is a finite set of data points sampled from the space. In such cases some of the most natural approaches to reconstructing the space begin with a process of building a combinatorial space out of the set of points. It is not always clear how to go about determining the global properties of such a space from the local data with which it is encoded. On the other hand, the subjects of differential topology and differential geometry are largely devoted to this problem, except that the spaces under investigation are not combinatorial, but rather smooth (and cannot in general be described with a finite amount of data). Some very general techniques have been developed over the recent decades for the study of smooth spaces. Forman's work is devoted to showing that many of these ideas can be adapted to apply to combinatorial spaces.