Abstract: We present a proof that the integral cohomology of a smooth toric variety is additively isomorphic to a torsion product involving the Stanley-Reisner ring of the fan defining the toric variety. Ingredients are a result of Gugenheim-May about the cohomology of pull-backs of principal torus bundles and a formality result for Davis-Januszkiewicz spaces.

Abstract: Given a surface S, the set of all homeomorphisms from S to itself which fix the boundary and preserve orientation form a group under composition; this is known as the group of homeomorphisms and is denoted Homeo+(S, ∂S). The mapping class group of S, denoted Mod(S), can be understood as Homeo+(S, ∂S) modulo homotopy. Mapping class groups are often studied using simple closed curves in the surface, that is, embeddings of the form f : S1 → S. More specifically, given a collection of simple closed curves in S, we can understand the behaviour of an element of the mapping class group f ∈ Mod(S) by observing what happens to the simple closed curves after applying a representative homeomorphism ϕ of the class f to the surface S. The curve graph, denoted C(S), is a graph whose vertices correspond to isotopy classes of essential simple closed curves in S and edges join vertices whose isotopy classes have disjoint representatives. In this talk, we will become familiarized with curves in surfaces in order to better understand mapping class groups of different surfaces. We will then discuss the nature of the curve graph and how it can be used as a combinatorial model of Mod(S).