Enumerative Real Algebraic Geometry

   Frank Sottile

   Consider the following hard problem:  Give meaningful information 
about the number r of real solutions to a system of multivariate 
polynomial equations.  Quite often much more is known than merely that 
r is between 0 and the number d of complex solutions.  This stronger
information is typically available when the system has some geometric 
structure.  Enumerative real algebraic geometry treats this hard problem
for systems coming from geometry.

   In this talk, I will survey some of what is known.  In particular,
I will discuss recent better understanding of the bounds of 0 and d.  
A picture is emerging: for systems from geometry, the upper bound
of d can always be obtained, and in many cases there are non-trivial 
lower bounds on the number of real solutions, with certain gaps that 
have been discovered experimentally.