Mathematics

In the first part of the talk I would like to recall basic information on the non-abelian Hirota-Miwa equation and on the corresponding map satisfying Zamolodchikov's tetrahedron condition. This includes the projective geometric interpretation of the Hirota map and of its multidimensional consistency, which points out towards a generalization of the map allowing for the quantum reduction. In the second part I will show that the Hermite-Pad\'e type I approximation problem leads in a natural way to Hirota's discrete KP system subject to an integrable constraint. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'e approximation problem is relevant. If time permits I will show how generalize this connection to the non-commutative level.