Conférencier: Kevin Purbhoo, Waterloo

Abstract: In the 1880's, Schubert solved a problem about enumerating subspaces of Cⁿ that have non-trivial intersections with other subspaces, beginning the study of what is now known as Schubert calculus. Nowadays, Schubert's problem is usually reinterpreted as a calculation in the cohomology ring of the Grassmannian, and this perspective has been vastly generalized. However, if we change Cⁿ to Rⁿ, the cohomology perspective no longer works and the situation is less well understood. In 1993, Boris and Michael Shapiro formulated a remarkable conjecture about real solutions to Schubert problems. It was proved in 2005 by Mukhin, Tarasov and Varchenko, using high powered machinery from quantum integrable systems. Since then, many applications and generalizations have been found or conjectured. In this talk, I will tell the story of the Shapiro-Shapiro conjecture, and then tell you about a new theorem (joint work with Jake Levinson), a not-so-obvious generalization. Our result is independent of Mukhin-Tarasov-Varchenko, implies the conjecture, and naturally lends itself to a much more intuitive proof.