Geometric Topology Seminar

Geometric Topology Seminar
Math Department of the Higher School of Economics, Room 209
Thursday, November 16, 14:00

Speaker: Mikhail Tyomkin (NRU Higher School of Economics)

Title: Barannikov's approach to Morse theory on manifolds with boundary

Abstract:

Given a Morse function on a closed manifold, one can construct a CW
complex by considering level sets, or a Morse chain complex by looking
at flow lines. The first one is unique, while the second one requires
additional data like a  Riemannian structure. However, it turns out
that a certain combinatorial structure on the set of critical points
can be well-defined and still depend only on the function (it is
called a Morse-Barannikov complex). A natural wish is to describe
bifurcations of such a structure when the function changes
one-parametrically and intersects a stratum of non-Morse functions.
This description allows one to attack a problem posed by Arnold: given
a germ of a function on 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.