Bastien Jean

Take a topological surface $S$ and $S_1$, $S_2$ two complex structures on $S$. There is no conformal isomorphism between $S_1$ and $S_2$ but we may ask what is the most nearly conformal homeomorphism between $S_1$, and $S_2$. This calls for a notion to measure conformality, this can be done by considering moduli of quadrilaterals and quasiconformal mapping. Those notions were introduced by Grötzsch in 1928. Those notions were later used by Ahlfors and Teichmüller and were the beggining of Teichmüller theory. I will explain those notions and how to use them to construct the Teichmuller distance and the Bers embedding of the Teichmüller space.