The inclusion-exclusion principle is a fundamental technique in combinatorics that allows passing between sizes of intersections and sizes of unions. We will explore a generalization of this formula where both the constituent pieces and the indexing set have extra structure, in particular a topology, inspired by the work of Vassiliev. We will then apply this technique to certain examples from algebraic geometry, namely the space of smooth polynomials and rational maps.

I will describe some joint work with Nick Salter in understanding the monodromy group associated to a complete linear system of curves in some smooth projective surface X. We prove that if X is simply connected and the linear system is sufficiently ample, then the monodromy group is of finite index in the mapping class group. I will try to be fairly concrete throughout the talk, in particular I will define the terms 'monodromy' and 'mapping class group'. I will also mostly focus on the concrete case of plane curves throughout the talk.