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A topological proof of the Shapiro-Shapiro Conjecture

Jake Levinson, University of Washington
Wednesday, January 29, 2020 - 3:30pm to 5:00pm
PDL C-401
Jake Levinson

The Wronski problem concerns Schubert problems involving tangent flags to the rational normal curve. Over the real numbers, this family has remarkable geometric and combinatorial structure: its entire topology can be explicitly described using Young tableaux and related combinatorial algorithms. Geometrically, the first hints of this structure were conjectured in the 1990s by Boris and Michael Shapiro, then proven in 2005 by Mukhin-Tarasov-Varchenko. Since then, the geometry and connection to combinatorics have been extended in many directions.

Most prior work has focused on the case where the tangent flags are all at real points on \$\mathbb{P}^1\$. In this case, the Wronski map is described by standard tableaux. I will describe a recent extension (joint with Kevin Purbhoo) that allows the osculation points to occur in complex conjugate pairs. We describe certain fibers of the map using domino tableaux, and we prove that the topological degree of the map is given by a character of the symmetric group.

Our result gives a new, topological proof of the original Shapiro-Shapiro conjecture. It also suggests some intriguing new topological-combinatorial questions to pursue.