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. 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.

I will lecture on these topics throughout the summer, revising the book I am writing and providing students with updated versions of each chapter. I expect at most 16-18 75-minute lectures.

The course will have three parts, grouped by geometry:
I) Overview (Chapter 1)
II) Toric varieties, upper and lower bounds (Chapters 2--6)
III) Grassmannians and the Shapiro Conjecture (Chapters 7--11)

Topics for each Chapter

1. Overview. Upper and lower bounds, Shapiro conjecture, and rational curves interpolating points in the plane.
2. Real solutions to univariate polynomials. Sturm's Theorem and the Fourier-Budan Theorem.
3. Sparse polynomial systems and toric varieties. Kouchnirenko's Theorem and Groebner degeneration.
4. Upper bounds. Descartes' rule of signs, Khovanski's fewnomial bound, bound for circuits.
5. Fewnomial upper bounds from Gale dual polynomial systems. Gale duality for polynomial systems, new upper bounds.
6. Lower bounds. Toric lower bound for sparse polynomial systems.
7. Enumerative real algebraic geometry.
     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
8. The Shapiro conjecture for Grassmannians.
     Wronski map, and derivation of Grassmannian version.
     Asymptotic form of the Shapiro Conjecture.
   
9. The Shapiro conjecture for Rational Functions.
     Rational functions with real critical points.
   
10. Proof of the Shapiro Conjecture.
    
11. Beyond the Shapiro Conjecture.
      Transversality, sums of squares, and discriminants.
      Maximally inflected curves.
      Shapiro conjecture for flag variety.
      Monotone and Secant conjectures.