Fixed point theorems and their applications / Ioannis Farmakis, Martin Moskowitz.

Includes bibliographical references (pages 219-227) and index.

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. 1.4. Fixed point theorems for groups of affine maps of [symbol] -- 2. Fixed point theorems in analysis. 2.1. The Schaüder-Tychonoff theorem. 2.2. Applications of the Schaüder-Tychonoff theorem. 2.3. The theorems of Hahn, Kakutani and Markov-Kakutani. 2.4. Amenable groups -- 3. The Lefschetz fixed point theorem. 3.1. The Lefschetz theorem for compact polyhedra. 3.2. The Lefschetz theorem for a compact manifold. 3.3. Proof of the Lefschetz theorem. 3.4. Some applications. 3.5. The Atiyah-Bott fixed point theorem -- 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 Poincaré recurrence theorem. 5.2. Symplectic geometry and its fixed point theorems. 5.3. Poincaré's last geometric theorem. 5.4. Automorphisms of Lie algebras. 5.5. Hyperbolic automorphisms of a manifold. 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. 6.5. Two conjugacy theorems for real linear Lie groups -- 7. Miscellaneous fixed point theorems. 7.1. Applications to number theory. 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.

This is the only book that deals comprehensively with fixed point theorems throughout mathematics. Their importance is due, as the book demonstrates, to their wide applicability. Beyond the first chapter, each of the other seven can be read independently of the others so the reader has much flexibility to follow his/her own interests. The book is written for graduate students and professional mathematicians and could be of interest to physicists, economists and engineers.

eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - Worldwide