The aim of this joint project is to investigate further the structure of gradient
flows of real analytic functions and particularly the role played by valleys, talwegs, and ridges of functions in the study of the flow at infinity. Near a critical line, the valley is formed by points at which the slope is minimal with respect to slopes measured at the same height. This object has played a pivotal role in the analysis of numerous dynamical systems having a gradient structure. This notion and its importance originally arose in the line of  Lojasiewicz works on gradient of tame functions and during the proof of Thom’s conjecture by Kurdyka-Mostowski/-Parusinski. Our goal is to study valleys and related notions into depth. Our principal conjecture roughly asserts that bounded gradient trajectories tend to line up with valleys for large time.