Tony Iarrobino: When do two nilpotent matrices commute?

The similarity class of an n by n nilpotent matrix B over a field k is given by its Jordan type, the partition P of n that specifies the sizes of the Jordan blocks. The variety N(B) parametrizing nilpotent matrices that commute with B is irreducible, so there is a partition Q=Q(P) that is the generic Jordan type for matrices A in N(B). The partition Q(P) has parts that differ pairwise by at least two, and Q(P) is stable: Q(Q(P))=Q(P). We discuss what is known about the map P to Q(P). A proof of a recursive conjecture by P. Oblak (2008), was recently announced by R. Basili after partial results by P. Oblak, T. Kosir, L. Khatami, and others. What is the set of partitions P having a given partition Q as maximum commuting orbit?  We prove a table Conjecture of P. Oblak and R. Zhao when Q has two parts, and state a ``Box Conjecture" in general for this set. This work is joint with Leila Khatami,  Bart van Steirteghem, and Rui Zhao.