Experimentation at the frontier of reality in Schubert calculus

   The story of the deeply surprising proof and consequences 
of the recent theorem of Mukhin, Tarasov, and Varchenko (n\'ee 
the Shapiro Conjecture) will be the subject of an address in 
this year's AMS Current Events Bulletin.  The interest in that 
conjecture was due in no small part to massive computational 
evidence that was amassed in its study, as the conjecture was 
originally considered too strong to possibly be true.

   This talk will be concerned with the gathering of evidence
for some extensions of the Shapiro conjecture.  In particular,
I will explain how our research team at Texas A&M University
is organizing and controlling a computational experiment involving 
hundreds of computers and hundreds of Gigahertz-years of 
computation in the pursuit of evidence for one generalization 
which we call the secant conjecture.