In the mid-1990s, Boris and Michael Shapiro conjectured that if f1, f2, ..., fd are linearly independent polynomials with complex coefficients such that their Wronskian has only real roots, then the complex linear subspace spanned by f1, f2, ..., fd has a basis consisting of polynomials with real coefficients. As stated, this conjecture is a problem in real algebraic geometry but it has several equivalent formulations that connect it to Schubert calculus and other areas of algebraic geometry and representation theory. Mukhin, Tarasov and Varchenko proved the Shapiro-Shapiro conjecture in 2005. Their proof was highly algebraic and offered little geometric insights.
In their paper, Levinson and Purbhoo formulate and prove a generalization of the Shapiro-Shapiro conjecture which addresses the case where the Wronskian has real coefficients (and not just real roots). Their proof, which uses geometric and topological methods, is conceptually simpler than the Mukhin-Tarasov-Varchenko proof, and suggests some intriguing connections between combinatorics and geometry.
