**Mon, August 31,** 3pm ET (8pm BST, 7am Tue NZST)**Tony Huynh**, Monash University**Subgraph densities in a surface**

**Youtube**

**Abstract:**We consider the following problem at the intersection of extremal and structural graph theory. Given a fixed graph $H$ and surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph that embeds in $\Sigma$? Here a

*copy*means a subgraph isomorphic to $H$. Exact answers are known for some $H$ when $\Sigma$ is the sphere. Our main result answers the question for all $H$ and $\Sigma$ (up to a constant factor). We show that the answer is $\Theta(n^{f(H)})$ where $f(H)$ is a graph invariant called the

*flap number*of $H$, which is independent of the surface $\Sigma$.

This is joint work with GwenaĆ«l Joret and David Wood.