Enumerative Real Algebraic Geometry
 Frank Sottile


   Enumerative Geometry is concerned with counting the number of geometric
figures satisfying some conditions imposed by some other, fixed figures.
For example, how many lines are tangent to 4 spheres?  Enumerative real
algebraic geometry is concerned with the real-number solutions to such 
problems, for example, how many of the solution figures can be real?   
Recent results in this area have, often as not, uncovered new and 
unexpected phenomena, and it is far from clear what to expect
in general.  Nevertheless, some themes are emerging.   My intention in 
this talk is to describe the current state of knowledge and indicate 
these themes.