Anton Alekseev:Eigenvalues, interlacing inequalities and tropical calculus

The same set of inequalities comes up in two seemingly different problems. The first setup is the interlacing inequalities satisfied by the (generalized) eigenvalues of an n by n Hermitian matrix. By the classical result of Guillemin-Sternberg, they define a completely integrable system (named after Gelfand-Zeitlin who discovered it in the context of Representation Theory of the unitary group). The second setup is the Boltzmann weights associated to (multi) paths on a planar network with n sources and n sinks.

In the talk, I explain the relation between the two theories. As an application, I give a new description of the Horn cone (spanned by eigenvalues of triples of Hermitian matrices adding up to zero).

This is a joint work with M. Podkopaeva and A. Szenes.