  This is an expository account of Katz's middle convolution operation on local
systems over ${\bf P}^1-\{q\_1,..., q\_n\}$. We describe the Betti and de Rham
versions, and point out that they give isomorphisms between different moduli
spaces of local systems, following V\"olklein, Dettweiler-Reiter,
Haraoka-Yokoyama. Kostov's program for applying the Katz algorithm is to say
that in the range where middle convolution no longer reduces the rank, one
should give a direct construction of local systems. This has been done by
Kostov and Crawley-Boevey. We describe here an alternative construction using
the notion of cyclotomic harmonic bundles: these are like variations of Hodge
structure except that the Hodge decomposition can go around in a circle.
