Finite rigid sets in curve complexes

In this talk, we will sketch a proof that curve complexes are {\em finitely rigid}. More concretely, for any orientable surface $S$ of finite topological type, we identify a finite subcomplex $X$ of the curve complex $C(S)$ such that every locally injective simplicial map $X\to C(S)$ is the restriction of an element of $Aut(C(S))$, unique up to the (finite) pointwise stabilizer of $X$ in $Aut(C(S))$. Furthermore, if $S$ is not a twice-punctured torus, then we can replace $Aut(C(S))$ in this statement with the extended mapping class group $Mod^\pm(S)$. This is joint work with
Christopher Leininger.