The Secant conjecture in the real 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.

   The Shapiro conjecture as it was originally stated is false 
for flag manifolds, but experimentation and theory have led to
what should be considered as the `correct' form for flag manifolds.
A proof of this generalization for certain two-step flag mainfolds
by Eremenko, Gabrielov, Shapiro, and Vainshtein led to
another generalization, which we call the Secant conjecture.
In this talk, I will explain some of this history, state
the Secant Conjecture, and describe some of the massive 
computational evidence (involving hundreds of Gigahertz-years of 
computation) in its favor.