Total positivity and integrable systems

Totally positive matrices were introduced in 1930 by Schoenberg in connection with the problem of estimating the number of real zeros of a polynomial. Since then, they have arised in connection with problems from different areas of pure and applied mathematics, including smnll vibrations of mechanical problems, statistics, approximation theory, combinatorics, graph theory, quantum field theories and integrability. In the seminar I shall review some classical theorems of total positivity and explain some applications to the theory of integrable systems.