Tame Flows

The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ : R×X → X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X . Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame. The typical tame gradient flow satisfies the Morse-Smale condition, and we prove that in the tame context, under certain spectral constraints, the Morse-Smale condition implies the fact that the stratification by unstable manifolds is Verdier and Whitney regular. We explain how to compute the Conley indices of isolated stationary points of tame flows in terms of their unstable varieties, and then give a complete classification of gradient like tame flows with finitely many stationary points. We use this technology to produce a Morse theory on posets generalizing R. Forman’s discrete Morse theory. Finally, we use the Harvey-Lawson finite volume flow technique to produce a homotopy between the DeRham complex of a smooth manifold and the simplicial chain complex associated to a triangulation. 2000 Mathematics Subject Classification. Primary 03C64, 06F30, 37B30, 58A07, 58A10, 58A17, 58A25, 58A35, 58E05, 58K50. Secondary. 55P05, 55U10, 57Q05, 57R05.