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The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro conjecture for the flag manifold and present massive computational experimentation in support of this refined conjecture. We also prove the conjecture in some special(More)
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 Grass-mannian 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(More)
We describe the setup, design, and execution of a computational experiment utilizing a supercomputer that is helping to formulate and test conjectures in the real Schubert calculus. Largely using machines in instructional computer labs during off-hours and University breaks, it consumed in excess of 350 GigaHertz-years of computing in its first six months(More)
  • 2007
The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the highest weight embedding of the ordinary Lagrangian Grassmannian, and one may study its defining ideal in this embedding. The Drinfel'd Lagrangian(More)
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