Experimentation in the Schubert Calculus

   The Schubert calculus provides a rich and 
well-structured class of problems in enumerative
algebraic geometry, which can be used as a 
laboratory for exploring ill-understood phenomena.
This lecture series will discuss two such phenomena,
reality of solutions to geometric problems and 
intrinsic structure as detected by Galois groups.
They will describe what we know and what we suspect,
and the large-scale computer experimentation that
has driven these questions and results.

Lecture 1: The Shapiro Conjecture and its proof.
  This lecture will cover the genesis, evidence
for, and resolution of the remarkable conjecture
of Boris and Michael Shapiro that all solutions to 
a Schubert problem on a Grassmannian are real when 
the flags giving the problem are tangent to a rational 
normal curve.

Lecture 2: Beyond the Shapiro Conjecture.
  While the Shapiro Conjecture may be formulated 
for any flag manifold, it typically fails in this 
generality.   In some cases these failures are quite 
interesting and the conjecture may be repaired.  
Also in some cases, if the tangent flags are replaced 
by disjoint secant flags, reality seems to hold.  
This lecture will survey the emerging landscape beyond 
the Shapiro Conjecture.

Lecture 3: Lower bounds and computing Schubert problems.
  Another phenomenon that was discovered en route to 
the proofs of the Shapiro Conjecture are lower bounds
on the number of real solutions to Schubert problems.
While experimentation reveals this to be wide-spread, 
this phenomenon is not understood well enough to formulate
a conjecture.  In addition to explaining these lower bounds,
this lecture will discuss how the massive (several teraHertz
years of computing) experiments studying these phenomena
are conducted.

Lecture 4: Intrinsic structure in the Schubert calculus.
   The symmetries of a geometric problem are encapsulated 
in its Galois group.  While Schubert problems are quite
structured, it is unknown which problems possess intrinsic
structure in that their Galois groups are not the full
symmetric group.  This lecture will discuss some preliminary
results and the different methods that are being deployed
to study Galois groups of the millions of computable Schubert
problems.