This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure set of zeroes of a general class of nonlinear operators. Appealing to a broad audience, it contains many important contributions to linear algebra, linear functional analysis, nonlinear functional analysis, and topology. The author gives several applications of the abstract theory to reaction diffusion equations and systems. The results presented cover a thirty-year period and cut across a variety of mathematical fields.

Introduction. Bifurcation from Simple Eigenvalues. First General Bifurcation Results. The Algebraic Multiplicity. Fundamental Properties of the Multiplicity. Global Bifurcation Theory. Some Applications. Bibliography.