Total Positivity and Real Schubert Calculus

This lecture titled "Total positivity and real Schubert calculus" is a Special Year Seminar presented by Steven Karp from the Institute for Advanced Study, scheduled for April 24, 2025. Explore the fascinating history of total positivity, beginning with its 1930s origins in totally positive matrices and their unique linear-algebraic and combinatorial properties. Discover modern applications to real-rooted polynomials, cluster algebras, and topological combinatorics in the first part of this 1-hour 50-minute presentation. The second part delves into recent research applying total positivity to Schubert calculus, which examines intersection problems among linear subspaces of C^n. Learn about the B. and M. Shapiro conjecture from the 1990s, which proposed that certain Schubert problems have all complex solutions being real, and how a totally positive version resolves conjectures by Sottile, Eremenko, Mukhin-Tarasov, and Karp himself. The presentation connects representation theory of symmetric groups, symmetric functions, and the KP hierarchy, featuring joint work with Kevin Purbhoo, Evgeny Mukhin, and Vitaly Tarasov. The seminar takes place at 10:00am in Simonyi 101.