Morse theory in the 1990's

Abstract

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.