Editing Monodromy (section)

{{Use American English|date = March 2019}}
{{Short description|Mathematical behavior near singularities}}
[[File:Imaginary log analytic continuation.png|thumb|The imaginary part of the [[complex logarithm]]. Trying to define the complex logarithm on <math>\C-\{0\}</math> gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of <math>\C-\{0\}</math> by a [[helicoid]] (an example of a [[Riemann surface]]).]]

In [[mathematics]], '''monodromy''' is the study of how objects from [[mathematical analysis]], [[algebraic topology]], [[algebraic geometry]] and [[differential geometry]] behave as they "run round" a [[Mathematical singularity|singularity]]. As the name implies, the fundamental meaning  of ''monodromy'' comes from "running round singly". It is closely associated with [[covering map]]s and their degeneration into [[Ramification (mathematics)|ramification]]; the aspect giving rise to monodromy phenomena is that certain [[function (mathematics)|function]]s we may wish to define fail to be ''single-valued'' as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a '''monodromy group''': a [[group (mathematics)|group]] of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called ''polydromy''.<ref name="König2015">{{cite book|last1=König|first1=Wolfgang|last2=Sprekels|first2=Jürgen|title=Karl Weierstraß (1815–1897): Aspekte seines Lebens und Werkes – Aspects of his Life and Work|date=2015|publisher=Springer-Verlag|isbn=9783658106195|pages=200–201|url=https://books.google.com/books?id=7IHDCgAAQBAJ&q=Karl+Weierstra%C3%9F+(1815%E2%80%931897)|access-date=5 October 2017|language=de}}</ref>