Total Positivity and its Applications

A matrix is called totally positive (resp. totally nonnegative)if all of its minors are positive (resp. nonnegative). This important class of matrices grew out of three separate applications: Vibrating systems, interpolation, and statistics. Since the pioneering work of Gantmacher/Krein, Schoenberg, and Karlin, the subject of total positivity has evolved into a prominent discipline in mathematics, and it continues to arise in numerous applications including: Weighted planar networks (ballot numbers); computer aided geometric design (shape preserving transformations); probability (moment matrices); and geometry (McMullen correspondence).

I intend to survey a number of current applications involving this class (including some mentioned above), and if time permits I will highlight some recent accomplishments connecting the eigenvalues of totally nonnegative matrices to the roots of certain biorthogonal polynomials.