Immersed Morse theory

We will consider an immersion of a manifold M into another manifold N. The k-th
stratum of N is the subset of points for which the inverse image in M consists of k
points. An immersed Morse function, roughly speaking, is a function from N to the
real numbers, that restricts to a Morse function on each stratum.
I will describe joint work with Maciej Borodzik, in which we develop this theory
analogously to classical Morse theory.
I will explain some of the applications we have gleaned thus far, as time permits.