"The project investigates the extrinsic geometry of foliations (expressed by the 2nd fundamental form of leaves) using the approach of the geometric flows.We propose to study not just a single foliated manifold (M, F, g), but rather – assuming a suitable local existence theory – a foliation with a one-parameter family of F-truncated metrics gt parametrized by a ‘time’ t.The research objectives are to develop Extrinsic Geometric Flow (EGF) on foliated manifolds recently introduced by the applicant (commonly with P. Walczak, arXiv:1003.1607v1) for codimension-one foliations as a new research tool for studying the extrinsic geometry of foliations. We shall also use the methods of geometric analysis and PDE’s, theory of Riemannian submanifolds, integral formulae for foliations, topology and dynamics of foliations, and computer simulations.The (hoped for) results concern(i) Development of EGF for a foliation of codimension-1 and of arbitrary codimension: existence and uniqueness theorems, converging as , behavior of curvature, singularities, extrinsic geometric solitons (for totally umbilical metrics, foliated surfaces and 3-manifolds), F-truncated variations of related total quantities and geometry of stable critical metrics etc;(ii) Applications of EGF to solutions of various problems (by Gluck-Ziller, Walczak, Toponogov etc) concerning the extrinsic geometry of foliations: minimizing functions like volume and energy defined for plane fields on Riemannian manifolds, the extrinsic Ricci and Newton transformation flows, foliated Riemannian submanifolds, combining integral formulae for real and complex foliations with the approach of EGF etc.The topic belongs to differential geometry and topology, subjects of pure mathematics."