Consider the following four questions: (a) What are the harmonic functions on a compact Riemannian Manifold? (Answer: constants) (b) What are the holomorphic functions on a compact complex manifold? (Answer: constants) (c) On a compact genus g surface, what is the dimension of the space of curl free and divergence free vector fields? (Answer: 2g) (d) What is the dimension of the space of holomorphic sections of the line bundle O(d)-->CP^1? (Answer: d+1) These all seem widely different and unrelated questions (except perhaps the fact that they all ask about the dimension of a certain vector space). Yet, they are all manifestations of a beautiful subject called Hodge Theory. It is the study of the enigmatic and ubiquitous object called the Laplacian. This talk will be an attempt to explain what is the Laplacain, how it arises and what insight it gives to questions in analysis, differential geometry and, if time permits, algebraic geometry.

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.