Monodromiematrix

The Monodromiematrix, also known as the monodromy matrix or monodromy group, is a mathematical object used in the study of differential equations and algebraic geometry. It describes how the solutions of a differential equation change when a parameter is varied along a closed loop. Specifically, if we consider a system of differential equations that depends on a parameter, and we move this parameter around a closed curve in its parameter space, the solutions will generally change. The monodromiematrix captures this change by representing it as a linear transformation of the solution space.

In the context of differential equations, particularly linear ones with regular singular points, the monodromy matrix relates the fundamental set of solutions at a point to itself after traversing a loop around a singularity. This matrix is crucial for understanding the behavior of solutions near these singular points. For instance, if the monodromy matrix is the identity, it implies that the solutions are single-valued as the parameter traverses the loop. If it is not the identity, the solutions are multi-valued.

In algebraic geometry, the monodromiematrix arises when studying families of algebraic varieties or mappings. It describes how the topological or geometric structure of these objects changes as a parameter varies. The study of the monodromiematrix provides insights into the fundamental properties and classifications of these mathematical structures. It is closely related to concepts like the Picard-Lefschetz theorem and the theory of vanishing cycles. The eigenvalues of the monodromiematrix are particularly important and are related to the exponents of the singularity.