# Tagged Questions

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: http://en.m....

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### What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
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### Is there a classification of isolated essential singularities?

In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked: Do we not care about essential singularities to classify ...
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### Need a hint for this integral

I'm trying to evaluate the following integral $$\int_0^{\infty} \frac{1}{x^{\frac{3}{2}}+1}\,dx.$$ This is an old complex analysis exam question, so I plan to use the residue theorem. How can I ...
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### How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
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### Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
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### Books or texts on singularity theory [closed]

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
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### $z_0$ non-removable singularity of $f\Rightarrow z_0$ essential singularity of $\exp(f)$

Let $z_0$ be a non-removable isolated singularity of $f$. Show that $z_0$ is then an essential singularity of $\exp(f)$. Hello, unfortunately I do not know how to proof that. To my opinion one ...
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### Does every open manifold admit a function without critical point?

Assume that $M$ is a non compact smooth manifold. Is there a smooth map $f:M\to \mathbb{R}$ such that $f$ has no critical point? The motivation comes from the conversations on this post.
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### Singularities of $e^{z - \frac{1}{z}}$

I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
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### Resolution of singularities of the determinant hypersurface

Let $$\det\nolimits_n=\sum_{\pi\in\mathfrak{S}_n} \operatorname{sgn}(\pi)\cdot x_{1,\pi(1)}\cdots x_{n,\pi(n)} \in \mathbb{C}[x_{ij}\mid 1\le i,j\le n]$$ be the determinant polynomial. It defines a ...
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### Distribution with singularities.

I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty \psi^{(n)}(n)$ is a distribution which has singularities of infinite order. Here $\psi$ is a test function ...
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### Singularities of $f(z)=z/\cos(z)$

Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not. I am a bit confused ...
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### Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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### Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
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### Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
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### How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
### Singularities of $\frac{\cos(z)}{(z-\frac{i}{2})^2}$ in $\mathbb{C}\cup\{\infty\}$
What are the (types of) singularities of $$f(z)=\frac{\cos(z)}{(z-\frac{i}{2})^2}$$ defined on $\mathbb{C}\cup\{\infty\}$? I know that $i/2$ is a pole of order $2$. There are no other singularities ...