Abstract
This thesis starts by constructing a complex line bundle for a simple 3-parameter family of 2 x 2 Hermitian matrices and explicitly computing its first Chern number. This example illustrates in the simplest possible context the connection between topology and degenerate eigenvalues of matrices. The central part of this thesis then follows. We present a geometric, vector bundle view of a large class of parameter-dependent boundary-value problems. In particular, we consider holomorphic families of linear ordinary differential equation systems on a finite interval which are subjected to prescribed parameter-dependent boundary conditions. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of lambda-dependent boundary-value problems, where lambda is a complex eigenvalue parameter. The fundamental analytical object of such lambda-dependent BVP's is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve that contains no eigenvalues of the problem can be characterized geometrically as the determinant of a transition function associated with the generalized Gardner-Jones bundle. The topology of this vector bundle, represented by its first Chern number, then yields precise information about the total number of eigenvalues of the problem in any prescribed subset of the complex lambda-plane. This result shows that the generalized Gardner-Jones bundle is an intrinsic geometric property of such lambda-dependent BVP's. The thesis then applies the generalized Gardner-Jones bundle framework to various examples, including one from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation. The final parts of the thesis contain exploratory attempts at understanding the geometric structure of some classes of parameter-dependent periodic linear systems and multiparameter linear systems. In the latter case, we explore curvature 2-forms. The thesis ends by summarising the achievements of this work, and discussing the various possible directions and objectives for future investigations.