I will give a gentle historical (and ongoing) account of matrix positivity and of operations that preserve it. This is a classical question studied for much of the past century, including by Schur, Polya-Szego, Schoenberg, Kahane, Loewner, and Rudin. It continues to be pursued actively, for both theoretical reasons as well as applications to high-dimensional covariance estimation. I will end with some recent joint work with Terence Tao (UCLA).
The entire talk should be accessible given a basic understanding of linear algebra/matrices and one-variable calculus. That said, I will occasionally insert technical details for the more advanced audience. For example: this journey connects many seemingly distant mathematical topics, from Schur (products and complements), to spheres and Gram matrices, to Toeplitz and Hankel matrices, to rank one updates and Rayleigh quotients, to Cauchy-Binet and Jacobi-Trudi identities, back full circle to Schur (polynomials).