Osculating Schubert calculus and twisted A1-degrees

Given a square system of real polynomials, we should never expect all solutions to the system to be real. Remarkably, a family of such solutions emerges in geometry from simple Schubert problems osculating rational normal curves (this is a deep and recent theorem previously known as the Shapiro-Shapiro conjecture). This has been a central topic in real algebraic geometry the past few decades, and reduces in many cases to the study of a map of real algebraic manifolds called the Wronski map. This map is not oriented in many cases, which resists our attempts to approach it through the lens of quadratically enriched enumerative geometry. We introduce the idea of twisted A1-degrees and A1-chambers, which allow us to repair conservation of number in many cases when orientability hypotheses may fail. We apply these ideas to the study of the Wronski map, exploring its twisted degrees, which unifies work from many authors. This is based on joint work with Jake Levinson.

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