The Shapiro Conjecture Frank Sottile Texas A&M University

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The Shapiro Conjecture

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Table of Contents
Abstract (IHP)     Abstract (Chambery)
Part I: The Wronski Map and Grassmannians

1. The Wronski Map
2. The Shapiro Conjecture
3. Rational Functions
4. Some Linear Algebra
5. Geometric Interpretation
6. Geometric Version of Shapiro Conjecture
7. Rational normal curve in 3-space
8. Three Points on the Rational Normal Curve
9. Three Lines Tangent to the Rational Normal Curve
10. Hyperboloid Through the Three Tangent Lines
11. The Fourth Tangent Line
12. Animation of the Fourth Tangent Line
13. Some Evidence for the Shapiro Conjecture
14. No Ramification ?
15. Transversality Conjecture
16. Discriminant Conjecture

Part II: Flag Manifolds and Beyond

17. Tangent and Secant Lines
18. The Secant Line
19. A Schematic
20. Secant Lines
21. Secant flags
22. Grassmannian Conditions on Flags
23. Monotone Conjecture
24. Some Evidence for the Monotone Conjecture
25. A Discriminant
26. Discriminant Conjecture
27. Bibliography