Barannikov's approach to Morse theory on manifolds with boundary

Given a Morse function on a closed manifold one can construct a CW complex by considering level sets or a chain Morse complex by looking at flow lines. The first one is not unique and the second one, on the other hand, requires additional data like Riemannian structure. However it turns out that a certain combinatorial structure on the set of critical points can be well-defined and still depend on the function only (it is called Morse-Barannikov complex). A natural wish to describe bifurcations of such structure while function changes one-parametrically and intersects a stratum of non-Morse functions arises. This description allows one to attack a problem posed by Arnold: given a germ of a function along the boundary of the manifold, estimate the number of critical points of its Morse continuation to the inside. This was done by Barannikov himself for the n-disk and by Pushkar in the general case. The talk will be purely elementary, no specific knowledge required.