The Conjecture of Shapiro and Shapiro

Table of Contents

1. Background.
   1. Real enumerative geometry.
   2. Linear systems.
   3. Pole placement problem.
   4. History of the conjectures.
2. Complete intersections: hypersurface Schubert conditions.
   1. Polynomial formulation of hypersurface Schubert conditions.
   2. The conjecture of Shapiro and Shapiro.
   3. The pole placement problem and geometry.
   4. Shapiro's conjecture and the pole placement problem.
   5. Equivalent systems of polynomials.
   6. Proof when (m,p)=(2,3).
   7. Computational evidence.
   8. Complexity of these computations.
   9. Why these equations are interesting.
3. General Schubert conditions and overdetermined systems.
   1. The Schubert calculus for the Grassmannian.
   2. The conjecture of Shapiro and Shapiro.
   3. Local coordinates for the intersection of 2 Schubert varieties.
   4. Equations for Schubert varieties.
   5. Proof in some cases.
   6. Computational evidence.
   7. Challenge problems.
4. Total positivity.
   1. Total positivity and the conjecture of Shapiro and Shapiro.
   2. Parameterization of totally positive matrices.
   3. Computational evidence.
5. Further remarks.
   1. A counterexample to the original conjecture of Shapiro and Shapiro.
   2. Further questions.
   3. Recent developments.
6. Acknowledgements.
7. Bibliography.