In the first two talks, we aim to prove the Dehn-Nielsen-Epstein-Baer theorem: Let $\Sigma_g$ denote the closed orientable connected surface of genus $g\geq1$ and $\text{Mod}^\pm (\Sigma_g)$ be extended mapping class group of $\Sigma_g$. Then there exists a canonical isomorphism $$\text{Mod}^\pm (\Sigma_g)\to \text{Out}\big(\pi_1(\Sigma_g)\big).$$ Injectivity of this isomorphism is more or less easy to prove, but the hardest part is surjectivity of this isomorphism, which requires topological rigidity of closed surfaces:
A homotopy equivalence between two closed orientable connected surfaces is homotopic to a homeomorphism.

One can ask the same question (rigidity problem) in the higher dimensions also (see below), and this is one of the fundamental questions in the (low/high dimensional) topology.
Borel Conjecture: Let $f\colon M\to N$ be a homotopy equivalence between two closed aspherical topological manifolds of the same dimension. Then $f$ is homotopic to a homeomorphism.