The Secant Conjecture in the real Schubert calculus

TL;DR

The paper introduces the Secant Conjecture, proposing that certain intersections in Grassmannians are always transverse with all points real when defined by secant flags along disjoint intervals, supported by theoretical and computational evidence.

Contribution

It formulates the Secant Conjecture as a generalization of the Shapiro Conjecture and provides both theoretical and extensive computational evidence for it.

Findings

The Secant Conjecture is supported by computational evidence from over one terahertz-year of calculations.

Observed phenomena in data suggest new insights into real intersections in Grassmannians.

Theoretical evidence aligns with computational results, strengthening the conjecture's validity.

Abstract

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real, if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for it as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some phenomena we observed in our data.