| Résume | Given k conjugacy classes of n by n invertible matrices, the Deligne-Simpson problem asks to determine whether or not there are matrices in the conjugacy classes whose product is the identity, and which are irreducible in the sense that the matrices have no common invariant subspace.
In 2004, I conjectured a necessary and sufficient condition in terms of the root system for a star-shaped graph, and in 2006 in joint work with Peter Shaw, we proved that the condition is sufficient.
After a long delay, the necessity of the condition has now been proved in joint work with Andrew Hubery, arxiv 2509.11998 (and independently by Cheng Shu, arxiv 2509.11841).
I shall discuss the joint work with Hubery, but I will also give an overview of my older work, which serves to introduce the relevant machinery: multiplicative preprojective algebras, reflection functors for them, weighted projective lines, parabolic bundles, compatible logarithmic connections and their monodromy.
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