Abstract: In this talk I will explore the notion of fold maps
which are a natural generalization of Morse functions. Morse functions
play a central role in the classification of manifolds and getting rid
of their critical points is a crucial step in the proof of the
h-cobordism theorem. I will describe a similar procedure for
eliminating so called fold-singularities. This is similar in spirit
but more flexible compared to the above-mentioned removal of critical
points as it allows to perform surgery on the underlying manifold. If
time permits I will also explain how this can be used to study fiber
bundles and their characteristic classes.