Abstract: To understand the Weyl group action on the corresponding real toric variety, it is necessary to understand that action on the coweight lattice modulo 2. We will generalize the construction of alcoves to get the orbit space and the size of each orbit.

Abstract: The curve graph of a surface encodes the intersection patterns of the isotopy classes of essential simple closed curves in the surface. In 1997, Ivanov showed that the curve graph can be used as a combinatorial model to study the extended mapping class group of a surface. Since then, with the objective of finding results analogous to that of Ivanov, several alternative versions of the curve graph have been defined and investigated in relation to certain algebraic invariants of a surface. For instance, Long–Margalit–Pham–Verberne–Yao successfully showed that the fine curve graph can be used as a combinatorial model for the group of homeomorphisms of a surface. In this expository talk, we will begin by introducing mapping class groups and several curve graphs associated to a surface. From here, we will investigate the relationship between curve graphs and homeomorphisms of a surface, stating several powerful theorems and even some preliminary results regarding separating curves. Finally, we will briefly discuss adjacent areas of research and the search for evidence supporting Ivanov’s infamous metaconjecture.