  This is a survey article on Morse theory based on lectures to graduate
students and advanced undergraduates. After a brief review of standard
material, mostly without proofs, the Morse theory of complex Grassmannian
manifolds is worked out in detail. In contrast to standard treatments, gradient
flow lines and their structure are emphasized. This leads to a convenient
interpretation of Schubert calculus via the momentum map of a torus action,
and, more generally, to a still-unfolding toric/combinatoric manifestation of
Morse theory. The latter provides a miniature version of the "field-theoretic
point of view" of Cohen-Jones-Segal, Betz-Cohen, and Fukaya.
