Total positivity and real Schubert calculus

In part 1, I will survey the history of total positivity, beginning in the 1930's with the introduction of totally positive matrices, which turn out to have surprising linear-algebraic and combinatorial properties. I will discuss some modern developments and applications to real-rooted polynomials, cluster algebras, topological combinatorics, and more.

In part 2, I will present recent work applying total positivity to Schubert calculus, which studies intersection problems among linear subspaces of C^n. In the 1990's, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. I will present a totally positive version of this result which resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself, based on connections with the representation theory of symmetric groups, symmetric functions, and the KP hierarchy. This is joint work with Kevin Purbhoo, and also with Evgeny Mukhin and Vitaly Tarasov.