9704360 Forman Robin Forman is investigating the relationship between differential topology and geometry, on the one hand, and combinatorial topology and geometry, on the other. He has already shown that many of the standard ingredients of differential topology and geometry, such as Morse theory, vector fields and their associated flows, and Ricci curvature and Bochner's method, have combinatorial analogs that can be applied to very general cell complexes. This has led to new results in combinatorial topology as well as to simplifications of some existing proofs. One current goal is to develop a combinatorial theory of vector bundles and connections that is suitable for studying both the classical question of combinatorial formulae for characteristic classes, as well as the new invariants of knots and 3- and 4-manifolds defined via the theory of smooth vector bundles and connections. Combinatorics is the study of finite sets. As a consequence, many problems in combinatorics lend themselves quite naturally to investigation via a computer. A typical problem in differential topology and geometry, on the other hand, seems to require an infinite amount of data. Forman's research is devoted to developing new methods of reducing such problems to combinatorics. In other words, he is developing new techniques to solve problems in topology and geometry that require keeping track of only a finite amount of data. In particular, these techniques enable one to investigate these problems via a computer. Moreover, his work has also proved useful in the opposite direction. Namely, he has shown that many of the standard techniques in differential topology and geometry can be adapted to apply to problems in combinatorics, an approach that has already proved useful in solving some of them. ***
