Preface Chapter 1. Overview 1.1. Introduction 1.2. Polyhedral bounds 1.3. Upper bounds 1.4. The Wronski map and the Shapiro Conjecture 1.5. Lower bounds Chapter 2. Real Solutions to Univariate Polynomials 2.1. Descartes's rule of signs 2.2. Sturm's Theorem 2.3. A topological proof of Sturm's Theorem Chapter 3. Sparse Polynomial Systems 3.1. Polyhedral bounds 3.2. Geometric interpretation of sparse polynomial systems 3.3. Proof of Kushnirenko's Theorem 3.4. Facial systems and degeneracies Chapter 4. Torie Degenerations and Kushnirenko's Theorem 4.1. Kushnirenko's Theorem for a simplex 4.2. Regular subdivisions and toric degenerations 4.3. Kushnirenko's Theorem via toric degenerations 4.4. Polynomial systems with only real solutions Chapter 5. Fewnomial Upper Bounds 5.1. Khovanskii's fewnomial bound 5.2. Kushnirenko's Conjecture 5.3. Systems supported on a circuit Chapter 6. Fewnomial Upper Bounds from Gale Dual Polynomial Systems 6.1. Gale duality for polynomial systems 6.2. New fewnomial bounds 6.3. Dense fewnomials Chapter 7. Lower Bounds for Sparse Polynomial Systems 7.1. Polynomial systems as fibers of maps 7.2. Orientability of real toric varieties 7.3. Degree from foldable triangulations 7.4. Open problems Chapter 8. Some Lower Bounds for Systems of Polynomials 8.1. Polynomial systems from posets 8.2. Sagbi degenerations 8.3. Incomparable chains, factoring polynomials, and gaps Chapter 9. Enumerative Real Algebraic Geometry 9.1. 3264 real conics 9.2. Some geometric problems 9.3. Schubert Calculus Chapter 10. The Shapiro Conjecture for Grassmannians 10.1. The Wronski map and Schubert Calculus 10.2. Asymptotic form of the Shapiro Conjecture 10.3. Grassmann duality Chapter 11. The Shapiro Conjecture for Rational Functions 11.1. Nets of rational functions 11.2. Schubert induction for rational functions and nets 11.3. Rational functions with prescribed coincidences Chapter 12. Proof of the Shapiro Conjecture for Grassmannians 12.1. Spaces of polynomials with given Wronskian 12.2. The Gaudin model 12.3. The Bethe Ansatz for the Gaudin model 12.4. Shapovalov form and the proof of the Shapiro Conjecture Chapter 13. Beyond the Shapiro Conjecture for the Grassmannian 13.1. Transversality and the Discriminant Conjecture 13.2. Maximally inflected curves 13.3. Degree of Wronski maps and beyond 13.4. The Secant Conjecture Chapter 14. The Shapiro Conjecture Beyond the Grassmannian 14.1. The Shapiro Conjecture for the orthogonal Grassmannian 14.2. The Shapiro Conjecture for the Lagrangian Grassmannian 14.3. The Shapiro Conjecture for flag manifolds 14.4. The Monotone Conjecture 14.5. The Monotone Secant Conjecture Bibliography Index of Notation Index