some guy on the street
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Geometric meaning of torsion in homotopy groups
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12 votes

Well, the silly answer is that $f:\mathbb{S}^k\to X$ represents a torsion element of order $p$ if $p \cdot f:\mathbb{S}^k\to X$ extends along $\mathbb{S}^k \hookrightarrow D^{k+1}$ to a map $\varphi: ...

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How should one present curl and divergence in an undergraduate multivariable calculus class?
11 votes

Depending (*) on the underlying degree of analyticity (**) in your calculus course, it might be just as well to start with the Stokes theorem, stating it as an existence and uniqueness theorem: ...

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Thom's Principle: rich structures are more numerous in low dimension
10 votes

Pro I think the examples given are instances of Guy's "strong law of small numbers". That seems at least poetic reason for low-dimensional specializations of your favorite theory to be different in ...

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A programming language that can only create algorithms with polynomial runtime?
8 votes

If I understand the paper's abstract, Yes.

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Subset of the plane that intersects every line exactly twice
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8 votes

By AC, choose a cardinal well-ordering of the lines in in the plane and any well-ordering of all the points. We proceed by transfinite induction. Suppose $A_l$ is a set of points, no three colinear, ...

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When is a Riemannian manifold an open subset of a complete one?
8 votes

If I understand you right, you're assuming that there is already a Riemann metric chosen for you. This of course integrates to a distance function ("metric" in the sense of metric spaces) whether or ...

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Clearing misconceptions: Defining "is a model of ZFC" in ZFC
6 votes

I suspect these sorts of problems arise in two ways: ignoring wrongness (e.g. Skolem's paradox, that there are countable models of set theory which believe in uncountable sets) and ignoring first-...

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What is the precise relationship between "prodsimplicial sets" and rooted trees?
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5 votes

There are a short list of operations described as generating the desired polyhedra: $ X : \mathrm{Prism} \vdash C X : \mathrm{Prism} $ $ l : \mathrm{list}\ \mathrm{Prism} \vdash \Pi l : \mathrm{...

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Why should I prefer bundles to (surjective) submersions?
5 votes

This is probably making a hash of the earlier answers, but bundles are special fibrations; specifically, they are fibrations with (not canonically) isomorphic fibers. And we all like fibrations, ...

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Side-Angle-Side Congruence and the Parallel Postulate
5 votes

On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point ...

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Theorems for nothing (and the proofs for free)
4 votes

Oh! From uniqueness of the countable dense linear order without endpoints: take (for instance) a countable ordinal $\lambda$, and consider the anti-lex order on $\mathbb{Q}\times\lambda$. This is a ...

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What do people mean by "subcategory"?
4 votes

Do people tend to mean the official definition? I think "official" belongs in scare-quotes... I tend to think that "subcategory" is an evil notion. I'm not published anywhere, but in my notes I ...

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Testing for Riemannian isometry
4 votes

Here's an irresponsible under-referenced response. On the one hand, this is hard! Indeed, I've read somewhere --- not handy --- that this question lurks among quantum gravity's many difficulties. ...

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Associativity with infinite nesting
3 votes

First, it's important that the infinite connect sum $A\# B\#A\#\cdots$ is not the limit of the finite connect sums $A,A\#B, A\#B\# A,\dots$; in fact, I'm sure the binary connect sum is as wrong a ...

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Conventional names for finite categories
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3 votes

I notice that the categories considered for naming here are all the domains, or shapes, of basic diagrams; an object, an arrow, an endomorphism (n.b., my instinct was just to call that $\mathbb{N}$), ...

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Tracking spectral sequence differentials
3 votes

Without meaning to be snarky, I think there's some confusion here about what spectral sequences are for. In particular, the sense in which a SpSeq (or even a long exact sequence!) serves as a ...

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What is your favorite isomorphism?
3 votes

I should say I'm fond of the Thom isomorphism, but I still find the contents rather mysterious.

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Efficiently sampling points uniformly from the surface of an n-sphere
3 votes

In case you don't want to worry about dividing by small norms (admittedly, less and less of an issue for higher $n$...) Generate a point $\mathbf{x}$ on the $(n-1)$-sphere; and generate a number $y\...

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A conceptual proof that local fibrations over paracompact spaces are global fibrations?
2 votes

The following runs out of steam towards the end; I may also be making important mistakes, so be on your guard --- but that's half the fun! Anyways, it was too long for a comment. Choose a locally-...

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Homology or cohomology?
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2 votes

(CW because it's more an over-long comment than a real answer.) I think there are too many competing normalizations to make a good choice. In lieu of sensible default, call one of them homology, and ...

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What is the earliest definition given by a universal mapping property?
2 votes

I'm betting on Supremum.

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Is there a category-theoretic definition of the arithmetic Grothendieck group
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2 votes

The classical group $K_0$ can also be thought of as consisting of equivalence classes of chain complexes of vector bundles, such that the exact sequences represent the zero of $K_0$ --- and ...

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How to motivate the skein relations?
2 votes

Why they are useful is related to one reason Polynomial invariants are useful themselves: they let you prove theorems, when they acutally do let you prove theorems. In particular, if you have a ...

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Do the empty set AND the entire set really need to be open?
2 votes

To address the question in a somewhat less-categorical way, I would point out that you can in fact do all of topology without using the expression "open set", by instead refering to the filter of ...

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Heuristic explanation of why we lose projectives in sheaves.
2 votes

Sorry if this is silly; but might it have something to do with needing in the sheaf category to consider the sheafified presheaf cokernel in order to talk about projections? that is, I (think I) can ...

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When and why do universal objects have extra properties?
2 votes

An incomplete answer on the subject of countable dense linear orders without endpoints; I left some other thoughts at the Cafe; on further reflection, one can think of the maps $\cdot\times \frac{p}{...

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Are all mathematical theorems necessarily true?
1 votes

As I understand the word "tautology" from Mathematical Logic; a logic $L$ means a formal system of deductions $\Sigma \vdash_L \Phi$, satisfying some rules, where $\Sigma$ is a set of propositions and ...

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Name for "lower/upper bounds" of arbitrary relations?
1 votes

In a pre-order $\prec$ (or a category) one can speak of initial objects $0$, or terminal objects $1$, meaning that $0\prec x$ for all $x$ --- (or $0\rightarrow^! x$ ) --- which also gives the notion ...

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When shorter means smaller?
1 votes

An incomplete answer; but perhaps it helps to rephrase the problem as below. The reason the round circle does have this property is that without loss of generality, the map $f$ fixes the origin; and ...

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Boundary of planar region
1 votes

Videtur I can't post comments of my own? This is not a complete answer. @buzzard, I'd say yours probably isn't a facetious comment, in that I can imagine a union of two Jordan curves --- that is, an ...

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