The complex projective spaces and Grassmannians have, in a sense, the nicest possible rational models. In the stable category, they are equivalent to a wedge of spheres. Unstably, they are formal, and also their underlying cohomology algebras are intrinsically formal as defined by Halperin and Stasheff. We prove analogous results in the equivariant case. The results feature in joint work with Vanny Doem, Chandal Nahak, and Soumyadip Thandar.

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.