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Experimentation at the frontiers of reality in Schubert calculus. (English) Zbl 1213.14117
Amdeberhan, Tewodros (ed.) et al., Gems in experimental mathematics. AMS special session on experimental mathematics, Washington, DC, January 5, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4869-2/pbk). Contemporary Mathematics 517, 365-380 (2010).

The Schubert calculus asks for the linear spaces that have specified positions with respect to other, fixed (flags of) linear spaces. The specified positions are a Schubert problem. The fixed linear spaces imposing the conditions give an instance of the Schubert problem. The number of solutions depends on the Schubert problem, while the solutions depend on the instance. The conjecture of Boris Shapiro and Michael Shapiro says that if the fixed linear space osculates a rational normal curve, then all solutions to the Schubert problem are real. The paper gives an overview about the state of the art concerning the conjecture. Based on ideas obtained by a lot of computer experiments the Shapiro conjecture was proved for Grassmannians [E. Mukhin, V. Tarasob, A. Varchenko, Ann. Math. (2) 170, No. 2, 863–881 (2009; Zbl 1213.14101)].

Computer experiments are described to test modified conjectures. One of them is the so–called secant conjecture. A flag is a secant along an interval I of a curve if every subspace in the flag is spanned by its intersections with I. The secant conjecture asserts that if the flags in a Schubert problem on a Grassmannian are disjoint in that they are secant along disjoint intervals of a rational normal curve, then every solution is real. A computer experiment is described verifying the secant conjecture in 4568553 instances using 4473 GigaHertz–years.

MSC:
14Q15Computational aspects of higher dimensional algebraic varieties
14N15Classical problems in algebraic geometry; Schubert calculus
Software:
Macaulay2; Sage; CoCoA