Lower bounds for real polynomial systems from sign-imbalanced posets.

  A new phenomenon in real algebraic geometry is the existence
of geometric problems with a provable lower bound on their number   
of real solutions.   If widespread, this would be enormously
important for applications.

  In this talk, I will discuss work with Evgenia Soprunova 
on a framework in which to understand these lower bounds.
In particular, we give polynomial systems associated to posets, 
and show that the sign-imbalance of the poset gives a lower 
bound on the number of real solutions to the associated system.
This uses combinatorics of toric varieties, toric degenerations, 
and some topology. Using sagbi degenerations, we recover results 
of Eremenko and Gabrielov on lower bounds in the Schubert
calculus.