# All Questions

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### Graphs such that contracting an edge decreases the chromatic number

Let $G = (V,E)$ be a finite, simple, undirected, connected graph, such that contracting an edge reduces the chromatic number. Does this imply that $G$ is complete?
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### Edge-colorings of plane graphs: do you know references where the following questions are studied?

Let $G$ be a plane graph (or more generally, a graph embedded on a surface) with a proper edge-coloring of $G$ with $k$ colors $\{1,\ldots,k\}$. I am interested in studying the cyclic permutations of ...
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### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...
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### “Immovable” topological spaces

Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$x \simeq_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$ Clearly ...
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### Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?

A function $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ is said to have the Darboux property whenever for all $X \subseteq \mathbf{N}$ and $y \in [0,f(X)]$, there exists $Y \subseteq X$ such that ...
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### Dimension of the singular locus of the moduli space of vector bundles

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with ...
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### “Interesting” projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?

The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would ...
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### Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion. What is the cardinality of ${\frak P}$, and what's ...
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### How the idea of adjoint matrix has been designed? [on hold]

I can understand the adjoint matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
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### Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension. Now I'm interested in a more specific fact about Kummer extensions. From Hooley's paper "On Artin's conjecture", we ...
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### Comparing spectrum of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators ...
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### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum $$\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,$$ where $k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$ is the degree of a Kummer extension for a ...
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### Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...
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### If two knots in $S^3$ are invertible cobordant (from both ends), are they equivalent?

Let $K_1,K_2$ be two knots in $S^3$ and assume that there exists a cobordism $(W;K_1,K_2)$ which is invertible from both ends. Does this imply that $K_1, K_2$ are equivalent? In the paper by D.W. ...