The University of Michigan Combinatorics Seminar


Abstract 

The talk will consist of two parts. The first one based on the joint work with S.Fallat and C.R.Johnson, deals with a problem of describing all possible determinantal inequalities for principal minors of totally nonnegative matrices. These inequalities generalize classical inequalities of Hadamard, Fischer, Szasz etc. I will describe all such inequalities of classical type, explain methods involved in complete solution of the problem for low dimensions and formulate a number of conjectures. In the second part (based on joint work with L.Faybusovich) results on nonstandard inverse problems for certain classes of Hessenberg and symmetric matrices will be presented. These inverse problems can be viewed as a generalization of both Hankel and Toeplitz moment problems and can be applied in a study of completely integrable lattices obtained as restrictions of full Toda flows to lowdimensional symplectic leaves.
Existence of a factorization into elementary bidiagonal factors
plays an important role in both parts of the talk. 