Combinatorics Colloquium: Positivity in real Schubert calculus
- Event Type
- Lecture
- Sponsor
- Combinatorics RTG group
- Location
- 245 Altgeld Hall
- Date
- Apr 4, 2024 3:00 - 3:55 pm
- Speaker
- Steven Karp (Notre Dame)
- Contact
- Alexander Yong
- ayong@illinois.edu
- Views
- 57
- Originating Calendar
- Combinatorics Research Area Calendar
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.