Real solutions to equations from geometry
  Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure. Particularly fruitful--both for information on real solutions and for applicability--are systems whose additional structure comes from geometry. Such equations from geometry for which we have information about their real solutions are the subject of these lecture notes, which focuses on bounds, both upper and lower, as well as situations in which all solutions can be real. 
Frank Sottile