Contents -- Preface and Acknowledgments -- Introduction -- 1 Early Fixed Point Theorems -- 1.1 The Picard-Banach Theorem -- 1.2 Vector Fields on Spheres -- 1.3 Proof of the Brouwer Theorem and Corollaries -- 1.3.1 A Counter Example -- 1.3.2 Applications of the Brouwer Theorem -- 1.3.3 The Perron-Frobenius Theorem -- 1.3.4 Google -- A Billion Dollar Fixed Point Theorem -- 1.4 Fixed Point Theorems for Groups of Affine Maps of Rn -- 1.4.1 Affine Maps and Actions -- 1.4.2 Affine Actions of Non Compact Groups -- 2 Fixed Point Theorems in Analysis -- 2.1 The Schauder-Tychonoff Theorem -- 2.1.1 Proof of the Schauder-Tychonoff Theorem -- 2.2 Applications of the Schauder-Tychonoff Theorem -- 2.3 The Theorems of Hahn, Kakutani and Markov-Kakutani -- 2.4 Amenable Groups -- 2.4.1 Amenable Groups -- 2.4.2 Structure of Connected Amenable Lie Groups -- 3 The Lefschetz Fixed Point Theorem -- 3.1 The Lefschetz Theorem for Compact Polyhedra -- 3.1.1 Projective Spaces -- 3.2 The Lefschetz Theorem for a Compact Manifold -- 3.2.1 Preliminaries from Differential Topology -- 3.2.2 Transversality -- 3.3 Proof of the Lefschetz Theorem -- 3.4 Some Applications -- 3.4.1 Maximal Tori in Compact Lie Groups -- 3.4.2 The Poincare-Hopf's Index Theorem -- 3.5 The Atiyah-Bott Fixed Point Theorem -- 3.5.1 The Case of the de Rham Complex -- 4 Fixed Point Theorems in Geometry -- 4.1 Some Generalities on Riemannian Manifolds -- 4.2 Hadamard Manifolds and Cartan's Theorem -- 4.3 Fixed Point Theorems for Compact Manifolds -- 5 Fixed Points of Volume Preserving Maps -- 5.1 The Poincare Recurrence Theorem -- 5.2 Symplectic Geometry and its Fixed Point Theorems -- 5.2.1 Introduction to Symplectic Geometry -- 5.2.2 Fixed Points of Symplectomorphisms -- 5.2.3 Arnold's Conjecture -- 5.3 Poincare's Last Geometric Theorem -- 5.4 Automorphisms of Lie Algebras.

5.5 Hyperbolic Automorphisms of a Manifold -- 5.5.1 The Case of a Torus -- 5.5.2 Anosov Diffeomorphisms -- 5.5.3 Nilmanifold Examples of Anosov Diffeomorphisms -- 5.6 The Lefschetz Zeta Function -- 6 Borel's Fixed Point Theorem in Algebraic Groups -- 6.1 Complete Varieties and Borel's Theorem -- 6.2 The Projective and Grassmann Spaces -- 6.3 Projective Varieties -- 6.4 Consequences of Borel's Fixed Point Theorem -- 6.5 Two Conjugacy Theorems for Real Linear Lie Groups -- 7 Miscellaneous Fixed Point Theorems -- 7.1 Applications to Number Theory -- 7.1.1 The Little Fermat Theorem -- 7.1.2 Fermat's Two Squares Theorem -- 7.2 Fixed Points in Group Theory -- 7.3 A Fixed Point Theorem in Complex Analysis -- 8 A Fixed Point Theorem in Set Theory -- Afterword -- Bibliography -- Index.