Positivity in real Schubert calculus, Steven Karp (Notre Dame) -- April 4th, 2024

Schubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2-dimensional subspaces of C^4 which intersect 4 given 2-dimensional subspaces nontrivially (it turns out there are 2 of them). 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. This conjecture inspired a lot of work in the area, including its proof by Mukhin-Tarasov-Varchenko in 2009. I will present a strengthening of this result which resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself, based on surprising connections with total positivity, the representation theory of symmetric groups, symmetric functions, and the KP hierarchy. This is joint work with Kevin Purbhoo.

The seminar co-organizers are Ian Cavey, Andrew Hardt, Shiliang Gao, Elizabeth Kelley, and Alexander Yong.