Informal Friday Seminar: Maxim -- Vanishing cycles and applications

In the first part, I will introduce vanishing cycles from a topological perspective, 
using the topology of complex hypersurface singularity germs as a motivation. The second 
part will be devoted to applications: (a) I plan to show how vanishing cycles can be 
used to compute the Euler characteristic of a complex projective hypersurface with 
arbitrary singularities; (b) I will briefly explain Beilinson’s gluing of perverse 
sheaves via vanishing cycles; (c) I will indicate how vanishing cycles can be used to 
categorify Donaldson-Thomas invariants; (d) If time permits, I will explain how motivic 
characteristic classes for vanishing cycles can be used in birational geometry to detect 
jumping coefficients of multiplier ideal and to characterize rational or du Bois 
singularities.