|Joint work with:
Chris Hillar (MSRI), Abraham Martin del Campo (Texas A&M),
James Ruffo (SUNY-Oneonta), Zach Teitler (Texas A&M), and
Frank Sottile (Texas A&M)
In 1993 Boris and Michael Shapiro made a remarkable and (at the time)
outlandish conjecture in the Schubert calculus, positing a way to ensure
that all solutions to a given problem were real. This Shapiro conjecture
became well-known due to large-scale computational experiments that
supported it. This led to a partial solution by Eremenko and Gabrielov,
and the full solution by Mukhin, Tarasov, and Varchenko in papers appearing
in the Annals of Mathematics.
The Shapiro Conjecture concerns flags that are tangent to the
rational normal curve. A proof of a special case of a generalization
of it suggested that it may hold if tangent flags were replaced by
secant flags. This Secant Conjecture has been the subject of a
large-scale computational experiment that is not only seeking evidence
for it, but also studying other, very subtle phenomena. Each instance
is tested by generating a polynomial system that models a Schubert
Calculus problem given by a choice of secant flags, and then determining
the numbers of real and complex solutions.
In this talk, I will describe the setup, design, and running of this project.
Largely using machines in instructional computer labs during off-hours and University
breaks, this experiment has consumed over 250 GigaHertz-years of computation,
testing over a billion instances of 274 different Schubert problems.
Some involve as many as 42 solutions in a ten-dimensional space and require 9 GigaHertz-hours
to compute a single instance. This experiment can serve as a model
for other large-scale mathematical investigations.