---
tag: C8HD62AB6Q
kind: concept
name: Monodromy
msc: [34M35, 34M03]
sources:
  - url: https://en.wikipedia.org/wiki/Monodromy
same_as:
  wikidata: Q543956
  wikipedia: Monodromy
---

## Summary

Monodromy describes how a multivalued object changes when it is continued around a loop.
Take a linear differential equation with singular points, and continue a basis of solutions around one of them.
The result is a new basis, related to the old one by an invertible matrix.
These matrices form a representation of the fundamental group of the plane minus the singular points, the monodromy representation.
It is a global invariant of the equation, and it is the subject of [[Hilbert's twenty-first problem|7RHW0KA5RY]].
See the [[definition|SK224Y1HNG]] and the related [[monodromy theorem|54QMKJF64A]].

## Explanations

### L1 human

The function $\sqrt{z}$ changes sign when one goes once around $0$.
The function $\log z$ gains $2 \pi i$.
These are the simplest cases of monodromy: walking around a singular point and coming back changes the value.
For a differential equation, the solutions are mixed among themselves in the same way, and a matrix records the mixing.