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Questions tagged [monodromy]

Monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity.

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What is a monodromy representation?

I'm currently studying the Global Torelli Theorem for K3 surfaces and I encountered the following section in Huybrechts book. If $\mathcal{X}\rightarrow S$ is a smooth proper morphism over a ...
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Does the monodromy action of a fiber bundle lie in the bundle's structure group?

Suppose we have a (topological) fiber bundle $p:E\to B$ with fiber $F$ and structure group $G$. Since $G$ acts on $F$ by homeomorphisms, it induces an action on the (integral) homology $H_*(F)$, i.e., ...
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The Monodromy and Parallel Form

According to famous Riemann-Hilbert correspondence, a flat connection gives a representation of fundamental group which is called monodromy. I would like to ask how does the monodromy relate to ...
ymm's user avatar
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Why monodromy theorem fails when we apply it to $\sqrt z$ or $logz$?

In my first complex analysis for physicists course I was introduced to monodromy theorem. The statement of this theorem should be the following: Given an open disk $U$ on the complex plane centered at ...
davise's user avatar
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Is it reasonable to evaluate the monodromy matrix of a periodic solution numerically with limits?

Suppose we are evaluating the stability of a periodic solution of a dynamic system: $$ \dot{x}=f(x,t) $$ where $x\in \mathbb{R}^n$, and $f$ being a smooth vector field periodic in $t$ with $f(x,t) = f(...
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monodromy representation associated with pushforward of constant sheaf

I am reading Geordie Williamson's guide to perverse sheaves and stuck on Example 5.11. Consider the map $f:\mathbb{C}^* \to \mathbb{C}^*: z \mapsto z^m$. Let $\underline{k}$ be the constant sheaf on $\...
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Extending holonomy/monodromy to finite spaces

I have encountered a situation that reminds me of holonomy/monodromy, but which takes place on a finite space. I am mostly looking for references, since this seems like a logical thing to investigate. ...
Kenneth Goodenough's user avatar
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Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)

Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead. Throughout, we will make use of the following result. Fact. For $H$ a Lie subgroup of $G$, there is a ...
Baylee Schutte's user avatar
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Evaluating roots of complex-valued functions [duplicate]

I have seen a lot of arguments recently where, for instance: $$ \sqrt{z^2-1} = \sqrt{z-1} \cdot \sqrt{z+1} \hspace{5mm} (*) $$ without ever specifying the chosen branch or the chosen branch cut (these ...
Matteo Menghini's user avatar
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Explicitly Finding a Monodromy of Hypergeometric Functions

In the book Higher Transcendental Functions, Volume I. McGraw-Hill. by Bateman, P.95, Bateman mentioned the two linearly independent solutions of a hypergeometric ODE, namely $u_1(z) = \enspace _2F_1(...
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How to get an action of topological fundamental group on the singular cohomology of a fiber?

Suppose $f:X\to Y$ is a proper smooth morphism of $\mathbb C$-varieties, and $y\in Y$ is a point. I want to get an action of $\pi_1(y,Y)$ (topological fundamental group) on $H^i_{sing}(X_y,\mathbb C)$....
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Which hyperbolic fibered knots have monodromy with a single singularity?

The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the ...
berto's user avatar
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Monodromy representation and base point

Let $S^1$ be a circle with a base point $o$. We know that there is an equivalence of categories between: (1) The category of local systems on $S^1$ (2) The category of finite dimensional ...
Mathstudent's user avatar
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Self contained exposition of second order Fuchsian ODEs

I am teaching a graduate course in Complex Analysis. I would like students to give an oral presentation at the end of the term on a topic which we did not cover in the lecture. So I am putting ...
Steven Gubkin's user avatar
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Local monodromy of Kummer Sheaves

Let $\mathbb{F}_q$ be a finite field, $\chi$ a complex (or $\overline{\mathbb{Q}_\ell}$) character of $\mathbb{F}_q^\times$, and $\mathcal{L}_\chi$ the Kummer sheaf on the multiplicative group $\...
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Expand $\cot z$ around $z_{0} = n\pi$

I'm trying to expand the function in a Laurent series $$f(z) = \cot z$$ around the singularity $$z_{0} = n\pi$$ Initially I tried expanding $\tan z$ with a Taylor expansion: $$\tan(z) = \tan(n\pi) + \...
Carlos Eduardo Staudt's user avatar
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Are there two isomorphic transitive subgroups of $\mathfrak{S}_n$ that are not conjugate?

I know that we can find two subgroups of $\mathfrak{S}_6$ both isomorphic to $\mathfrak{S}_5$ that are however not conjugate (here) in $\mathfrak{S}_6$. These subgroups are not conjugate precisely ...
Antoine de La Humâlerie's user avatar
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Laurent expansion of Meijer's G function

I am considering the following equation (a Generalized hyper-geometric equation): $$\left(D-\beta_1\right)\left(D-\beta_2\right)f(x)-x\left(D+1-\alpha_1\right)\left(D+1-\alpha_2\right)f(x)=0$$ where ...
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The Monodromy Representation of a Riemann Riemann Surface with a Singular Projective Structure

Let $R$ be a Riemann Surface with a Regular Projective Structure $\nu$. Questions: What is the Monodromy Representation $\mu: \pi_1(R) \longrightarrow PSL(2,\mathbb{C})$, how to define it? If the ...
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Understanding some concepts about Monodromy

I'm reading Simpson's paper "Higgs Bundles and Local Systems". There he defines for a local system $V$ of vector spaces on a compact Kahler manifold $M$ we have a representation of ...
Angry_Math_Person's user avatar
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How to Define the Monodromy Action on the Universal Cover

Let $M$ be a connected manifold, and let $\rho : \tilde{M} \to M$ be its universal covering. Fix $x_0 \in M$. I know how to define the $\pi_1(M,x_0)$-action on the fiber $\pi^{-1}[\{x_0\}] \subseteq \...
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Necessary & Sufficient condition for the existence of Analytic Continuation

While solving the problems on Analytic continuation from Gamelin's book; I encountered this one- still unsolved: Let $D= \{0 < |z| < \epsilon\}$ and suppose $f$ is holomorphic at $z_{0} \in D$ ...
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Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\\\\ \frac1{\sqrt{-1}} &= \frac1i \\\\ \frac{\sqrt1}{\sqrt{-1}} &...
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