Understanding, finding, or even deciding 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 will be the subject of this short course.

    We will focus on equations from toric varieties and homogeneous spaces, particularly Grassmannians. Not only is much known in these cases, but they encompass some of the most common applications. The results we discuss may be grouped into three themes:
  (1) Upper bounds on the number of real solutions.
  (2) Geometric problems that can have all solutions be real.
  (3) Lower bounds on the number of real solutions
Upper bounds as in (1) bound the complexity of the set of real solutions---they are one of the sources for the theory of o-minimal structures which are an important topic in this trimester. Lower bounds as in (3) give an existence proof for real solutions. Their most spectacular manifestation is the non-triviality of the Welschinger invariant, which was computed via tropical geometry. This is also explained in other courses this trimester at the Centre Borel.

The course will have three parts, grouped by geometry:
I) Overview (Lecture 1)
II) Toric Varieties (Lectures 2--4)
III) Grassmannians (Lectures 5--8)

Topics for each lecture.

1. Overview. Upper and lower bounds, Shapiro conjecture, and rational curves interpolating points in the plane.
2. Sparse polynomial systems and toric varieties. Kouchnirenko's Theorem and Groebner degeneration.
3. Upper bounds. Descartes' rule of signs, Khovanski's fewnomial bound, bound for circuits.
4. Lower bounds. Toric lower bound for sparse polynomial systems.
5. Enumerative real algebraic geometry.
     Recall Sturmfels's Theorem.
     Problem of plane conics, four balls, and the Stewart platform.
     Schubert calculus for the Grassmannian.
     Reality in the Schubert calculus and Vakil's Theorem.
     Reality in the quantum Schubert calculus.
     Statement of Shapiro Conjecture
6. The Shapiro conjecture for Grassmannians.
     Wronski map, and derivation of Grassmannian version.
     Prove Asymptotic result.
   
7. The Shapiro conjecture for Curves.
     Rational functions with real critical points.   Maximally inflected curves (topological restrictions and open problems).
   
8. Beyond the Shapiro Conjecture (A selection of the topics below).
     Discuss computations.
     Shapiro for simple Schubert calculus implies general Shapiro.
     Transversality, sums of squares, and discriminants.   Conjecture makes sense for any flag variety.
     Simple counterexample.
     Monotone conjecture.
     Experimental evidence and show some tables.
     Proof by Eremenko, Gabrielov, Shapiro, and Vainshtein.
     Extension to Secant flags.