Foreword -- Introduction -- Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals -- 1. Introduction -- 2. Real closed fields -- 2.1. Definition and first examples of real closed fields -- 2.2. Cauchy index and real root counting -- 3. Real root counting -- 3.1. Sylvester sequence -- 3.2. Subresultants and remainders -- 3.3. Sylvester-Habicht sequence -- 3.4. Quadratic forms, Hankel matrices and real roots -- 3.5. Summary and discussion -- 4. Complexity of algorithms -- 5. Sign determinations -- 5.1. Simultaneous inequalities -- 5.2. Thom's lemma and its consequences -- 6. Existential theory of reals -- 6.1. Solving multivariate polynomial systems -- 6.2. Some real algebraic geometry -- 6.3. Finding points on hypersurfaces -- 6.4. Finding non empty sign conditions -- References -- Nash functions and manifolds -- 1. Introduction -- 2. Nash functions -- 3. Approximation Theorem -- 4. Nash manifolds -- 5. Sheaf theory of Nash function germs -- 6. Nash groups -- References -- Approximation theorems in real analytic and algebraic geometry -- Introduction -- I. The analytic case -- 1. The Whitney topology for sections of a sheaf -- 2. A Whitney approximation theorem -- 3. Approximation for sections of a sheaf -- 4. Approximation for sheaf homomorphisms -- II. The algebraic case -- 5. Preliminaries on real algebraic varieties -- 6. A- and B-coherent sheaves -- 7. The approximation theorems in the algebraic case -- III. Algebraic and analytic bundles -- 8. Duality theory -- 9. Strongly algebraic vector bundles -- 10. Approximation for sections of vector bundles -- References -- Real abelian varieties and real algebraic curves -- Introduction -- 1. Generalities on complex tori -- 1.1. Complex tori -- 1.2. Homology and cohomology of tori -- 1.3. Morphisms of complex tori.

1.4. The Albanese and the Picard variety -- 1.5. Line bundles on complex tori -- 1.6. Polarizations -- 1.7. Riemann's bilinear relations and moduli spaces -- 2. Real structures -- 2.1. Definition of real structures -- 2.2. Real models -- 2.3. The action of conjugation on functions and forms -- 2.4. The action of conjugation on cohomology -- 2.5. A theorem of Comessatti -- 2.6. Group cohomology -- 2.7. The action of conjugation on the Albanese variety and the Picard group -- 2.8. Period matrices in pseudonormal form and the Albanese map -- 3. Real abelian varieties -- 3.1. Real structures on complex tori -- 3.2. Equivalence classes for real structures on complex tori -- 3.3. Line bundles on complex tori with a real structure -- 3.4. Riemann bilinear relations for principally polarized real varieties -- 3.5. Moduli spaces of principally polarized real abelian varieties -- 3.6. Real theta functions -- 4. Applications to real curves -- 4.1. The Jacobian of a real curve -- 4.2. Real theta-characteristics -- 4.3. Examples -- 4.4. Moduli spaces and the theorem of Torelli -- 4.5. Singular curves -- References -- Appendix -- Mario Raimondo's contributions to real geometry -- Mario Raimondo's contributions to computer algebra.