Abstract: The Horn problem is a classical problem of Linear Algebra which establishes a complete set of inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. It was solved by Klyachko and Knutson-Tao in the end of 1990s.

Surprizingly, the same set of inequalities comes up in the problem of maximal multi-paths in a planar network equipped with Boltzmann weights on its edges. The link between the two problems is via the tropical limit. In order to control this limit we are using the Liouville volume on the space of solutions of the Horn problem.

The talk is based on a joint work with M. Podkopaeva and A. Szenes.