Any smooth function on a compact manifold has at least 2 critical points: maximum and minimum. The celebrated Morse inequalities imply that on most manifolds each smooth function has many more critical points. For example, any function on the two-dimensional torus has at least 4 critical points (provided those points are non-degenerate).
In 1981, S. Novikov extended the Morse inequalities to
multi-valued functions. In my talk, I'll review the
Morse and Novikov theories and present a generalization
of the Novikov inequalities to multi-valued functions
with non-isolated critical points due to M. Farber and
myself. If the time permits, I'll also discuss some