Abstract:

This talk provides a gentle introduction to totally positive matrices and Polya frequency sequences. We will see basic examples, history, and fundamental results on total positivity, variation diminution, and sign non-reversal – as well as a few proofs to illustrate how the main ingredients fit together. Several classical results (and one

Hypothesis) from before 1955 feature in this journey. We will end by connecting Polya frequency sequences to the Laguerre–Polya class and hence to Polya–Schur multipliers, and mention 21st century incarnations of the latter.