Singularities old and new, I

In the early days of glory, geometric topology scored many astonishing achievements in the theory of manifolds and submanifolds, a few samples of which will be mentioned. In many situations singularities are unavoidable, in topology and in other domains where one wants to apply topological methods, such as algebraic geometry, combinatorics, physics. We will review intersection homology and certain characteristic classes of stratified spaces, and discuss lattice points in polytopes as a sample application. If there is time, a little F-theory (elliptic fibrations) from physics will also be mentioned. We will then conclude with work of Kjuchukova and others on the application and potential application of singularities to the theory of low-dimensional manifolds.