Hofer Norms on Braid Groups and Quantitative Heegaard-Floer Homology

Given a Lagrangian link with k components in the disc (or k+g components on a surface with non-empty boundary and genus g), it is possible to define an associated Hofer norm on the braid group with k strands. Starting from basic definitions such as those of Hofer metrics, braids and Lagrangian links, we are going to give a sketch of the proof of this result. The main tool is going to be Quantitative Heegaard-Floer Homology (as constructed by Cristofaro-Gardiner, Humilière, Mak, Seyfaddini and Smith), which gives rise to families of quasimorphisms detecting linking numbers of braids. This talk is based on my own work for the case of the disc, and on a joint work with Ibrahim Trifa for the extension to higher genus.