Maya Sankar, The Turán Numbers of Homeomorphs

Let $X$ be a 2-dimensional simplicial complex. Denote by ${\text{ex}}_{\mathrm{hom}}(n,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of ${\text{ex}}_{\mathrm{hom}}(n,X)$ have recently been determined for all surfaces $X$. I will discuss these results, as well as ongoing work bounding ${\text{ex}}_{\mathrm{hom}}(n,X)$ for arbitrary 2-dimensional simplicial complexes $X$.