Arnold's seminar February 22, 2000 P. Pushkar' "Morse-type inequalities on a manifold with boundary" Let M be a manifold with boundary dM and f, a generic smooth function defined in a neighbourhood of dM. Let g be an arbitrary prolongation of f on M. Question: what is the minimal number of critical points that g must have? Example 1: Morse theory (dM is empty). Example 2: M=[0,1] and f near the endpoints behaves like the following function: * * * * * * * * * Then any extension g of f must have at least 2 critical points on M. Example 3: we will develop a homological algebra to treat the general case. The triple (M,dM,f) gives rise to a combinatorial graph G such that the lower bound for the number of critical points of g is given by #(vertices of G) - 2*(max number of edges without common vertices).