Illustration of analytic continuation along a curve (only a finite number of the disks are shown).Analytic continuation along a curve of the natural logarithm (the imaginary part of the logarithm is shown only).
In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function) along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued.
Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties.
In complex analysis, the monodromytheorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea...
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave...
continuation for these functions beyond the interior of the unit circle. The monodromytheorem gives a sufficient condition for the existence of a direct analytic...
calculus List of complex analysis topics Monodromytheorem Real analysis Riemann–Roch theorem Runge's theorem "Industrial Applications of Complex Analysis"...
residue characteristic of K is different from ℓ, Grothendieck's ℓ-adic monodromytheorem sets up a bijection between ℓ-adic representations of WK (over Qℓ)...
a version of the monodromytheorem for coverings. It has been generalized to give proofs of the more general Poincaré polygon theorem. (Note that the special...
is a Deligne conjecture on monodromy, also known as the weight monodromy conjecture, or purity conjecture for the monodromy filtration. There is a Deligne...
fundamental group. In other words, the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the...
defined away from 0,1,∞. Consider the function z → ω(f(z)). By the Monodromytheorem this is holomorphic and maps the complex plane C to the upper half...
Thurston's geometrization theorem in this special case states that if M is a 3-manifold that fibers over the circle and whose monodromy is a pseudo-Anosov diffeomorphism...
Nicholas M. Katz (January 1970). "Nilpotent connections and the monodromytheorem: Applications of a result of Turrittin". Publications Mathématiques...
1 ( 0 ) ϕ ( T ) {\displaystyle \phi ^{-1}(0)\phi (T)} is known as the monodromy matrix. In addition, for each matrix B {\displaystyle B} (possibly complex)...
the integral of closed 1-forms to be extended to continuous paths: Monodromytheorem. If ω is a closed 1-form, the integral ∫γ ω can be extended to any...
methods. He introduced the Katz–Lang finiteness theorem. Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematical Studies, Princeton...
transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has...
"Connections, curvature, and p-curvature", preprint. Katz, N., "Nilpotent connections and the monodromytheorem", IHES Publ. Math. 39 (1970) 175–232....
of p. The monodromy group acts by permuting the factors, and thus forms the monodromy representation of the Galois group of p. (The monodromy action on...
263 (Ex. 30.10.1). Bloch, Spencer; Masha, Vlasenko. "Gamma functions, monodromy and Apéry constants" (PDF). University of Chicago (Paper). pp. 1–34. S2CID 126076513...
\{0,1,\infty \}} . Here the monodromy around 0 and 1 can be computed using Picard–Lefschetz theory, giving the monodromy around ∞ {\displaystyle \infty...
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map,...
first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop...
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...
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