We consider the derived categories of modules over a certain family A_m of graded
rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which
are the Milnor fibres of simple singularities of type A_m. We show that each of these two
rather different objects encodes the topology of curves on an (m+1)-punctured disc. We
prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules
and that it injects into the symplectic mapping class group of Milnor fibers. The
philosophy behind our results is as follows. Using Floer cohomology, one should be able to
associate to the Milnor fibre a triangulated category (its construction has not been
carried out in detail). This triangulated category should contain a full subcategory which
is equivalent, up to a slight difference in the grading, to the derived category of
A_m-modules. The full embedding would connect the two occurrences of the braid group, thus
explaining the similarity between them.