Fixed point theory concerns itself with a very simple, and basic, mathematical setting. For a functionf that has a setX as bothdomain and range, a ?xed point off isa pointx ofX for whichf(x)=x. Two fundamental theorems concerning ?xed points are those of Banach and of Brouwer. In Banach's theorem, X is a complete metric space with metricd andf:X?X is required to be a contraction, that is, there must existL< 1 such thatd(f(x),f(y))?Ld(x,y) for allx,y?X. Theconclusion is thatf has a ?xed point, in fact exactly one of them. Brouwer'stheorem requiresX to betheclosed unit ball in a Euclidean space and f:X?X to be a map, that is, a continuous function. Again we can conclude that f has a ?xed point. But in this case the set of?xed points need not be a single point, in fact every closed nonempty subset of the unit ball is the ?xed point set for some map. ThemetriconX in Banach'stheorem is used in the crucialhypothesis about the function, that it is a contraction. The unit ball in Euclidean space is also metric, and the metric topology determines the continuity of the function, but the focus of Brouwer's theorem is on topological characteristics of the unit ball, in particular that it is a contractible ?nite polyhedron. The theorems of Banach and Brouwer illustrate the di?erence between the two principal branches of ?xed point theory: metric ?xed point theory and topological ?xed point theory.

Table of Contents:
Homological Methods in Fixed Point Theory.- Coincidence Theory.- On the Lefschetz Fixed Point Theorem.- Linearizations for Maps of Nilmanifolds and Solvmanifolds.- Homotopy Minimal Periods.- Periodic Points and Braid Theory.- Fixed Point Theory of Multivalued Weighted Maps.- Fixed Point Theory for Homogeneous Spaces A Brief Survey.- Equivariant Fixed Point Theory.- A Note on Equivariant Fixed Point Theory.- Equivariant Degree.- Bifurcations of Solutions of SO(2)-Symmetric Nonlinear Problems with Variational Structure.- Nielsen Theory.- Nielsen Root Theory.- More about Nielsen Theories and Their Applications.- Algebraic Techniques for Calculating the Nielsen Number on Hyperbolic Surfaces.- Fibre Techniques in Nielsen Theory Calculations.- Wecken Theorem for Fixed and Periodic Points.- A Primer of Nielsen Fixed Point Theory.- Nielsen Fixed Point Theory on Surfaces.- Relative Nielsen Theory.- Applications.- Applicable Fixed Point Principles.- The Fixed Point Index of the Poincaré Translation Operator on Differentiable Manifolds.- On the Existence of Equilibria and Fixed Points of Maps under Constraints.- Topological Fixed Point Theory and Nonlinear Differential Equations.- Fixed Point Results Based on the Wa?ewski Method.