Konstanze Rietsch King's College, London Toeplitz matrices, total positivity, and quantum cohomology. Friday 6th, December 12:05-12:55pm, Carslaw 373. A matrix is called totally positive if all of its minors are positive. Total positivity for infinite by infinite Toeplitz matrices was first studied in the 1950's by Schoenberg and collaborators, and has close connections with Schur functions and representations of the infinite symmetric group. The aim of this talk is to describe the set of totally positive unitriangular Toeplitz matrices in $GL_n$ and explain a somewhat surprising relationship with quantum cohomology rings of flag varieties and quantum Schubert polynomials.