soft question - A single paper everyone should read?

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.

Do you have such an example?

Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.

And as usual let's do one per post and vote freely, vote a lot.

galactic dynamics - What is $v_{GSR}$

The $v_{GSR}$ in the linked paper seems to be the radial velocity (RV, in $km.s^{-1}$) with respect to the GSR. This information is mentioned in the Abstract section.

I don't know exactly what kind of coordinates you have and how exactly are your velocities expressed (Cartesian vectors?), but what you probably need to do is to convert the velocity vector of a star into the radial component using the formula

$r=sqrt{x^2 + y^2}$

You need to make sure where the origin of your coordinates is.

This link clarifies what GSR is. Basically there are several frames of reference that are used on the galactic scales. LSR (Local Standard of Rest) is co-rotating with the gas and dust of the galaxy (so an average star is at rest in LSR), while GSR is not rotating with the galaxy and can be calculated from LSR using a formula

$v_{GSR}=v_{LSR}+220sin(l)cos(b)$

where l and b are the Galactic longitude and latitude.

ag.algebraic geometry - Can there exist two non-equivalent equivariant actions of a group on vector bundle?

Maybe I am miss understanding the question, but it seems the answer is yes.

Take your favorite G-space, mine is $S^1$ with the $mathbb{Z}/2$-action "flip". Then consider the trivial vector bundles $S^1 times V$, where $V$ is a $G$-representation. In my favorite example $V = mathbb{R}$ can be either the trivial representation or the sign representation. Taking the diagonal $G$-action gives an equivariant action of the group on the vector bundle. They are distinct for distinct representations (at least in the $S^1$-example) yet the underlying vector bundles are the same if the representations have the same dimension (they are trivial bundles after all).

co.combinatorics - Generalizations of the Birkhoff-von Neumann Theorem

I am cheating a little to give this answer, because I am fairly sure that it is part of Gil's motivation in asking the question. The most natural generalization of the Birkhoff hypothesis to quantum probability is only true for qubits. (It might also be true for a qubit tensor a classical system; I did not check that case.)

A quantum measurable space is a von Neumann algebra. We are most interested in the finite-dimensional case, where classically "measurable space" is just a fancy name for the random variables on a finite set. A finite-dimensional von Neumann algebra is a direct sum of matrix algebras. In particular, $M_2$ is called a qubit and $M_d$ is called a qudit.

To make a long story short, the Birkhoff hypothesis can be stated for a direct sum of $a$ copies of $M_b$, or $aM_b$. In this setting, a doubly stochastic map $E$ is a linear map from $aM_b$ to itself that preserves trace, that preserves the identity element, and that is completely positive. In this setting, $E$ is completely positive if it takes positive semidefinite elements of $aM_b$ to positive semidefinite elements, and if $E otimes I$ also has that property on the algebra $aM_b otimes N$ for another von Neumann or $C^*$-algebra $N$. The natural analogue of permutation matrices are the *-algebra automorphisms of $aM_b$. These are permutations of the matrix blocks, composed with maps of the form $E(x) = uxu^*$, where $u$ is a unitary element of $aM_b$. The question as before is whether the doubly stochastic maps are the convex hull of the automorphisms.

This Birkhoff hypothesis is true for $M_2$, false for $M_d$ for $d ge 3$, and I should check it for $nM_2$. It is true for $aM_1 = amathbb{C}$, because then it is the usual Birkhoff-von Neumann theorem.

I am left wondering about two infinite classical versions of Birkhoff's theorem, for the algebras $ell^infty(mathbb{N})$ and $L^infty([0,1])$. In the former case, one would ask whether any stochastic map that preserves counting measure (even though counting measure is not normalized) is an infinite convex sum of permutations of $mathbb{N}$. In the latter case, whether any stochastic map that preserve Lebesgue measure is a convex integral of measure-preserving permutations of $[0,1]$. Addendum: At least the discrete infinite case is addressed, with generally positive results, in this review and in this older review. The older paper also raises the continuous question but with no results. However, with some more Googling I found this counterexample paper.

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Since Gil asks for a reference, a recent one is Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem, by Mendl and Wolf.

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Here also is a more orthodox combinatorial generalization of the Birkhoff theorem, and also another case that I once encountered that is between a generalization and a non-generalization. Since Gil now offers a bounty, maybe it's better to merge this answer with the other one.

A doubly stochastic matrix can be interpreted as a flow through a directed graph, with unit capacities. (See Unimodular matrix in Wikipedia; I learned about this long ago from Jesus de Loera.) Any such graph has a polytope of flows, called a network flow polytope. Any network flow polytope has integer vertices, because it is a totally unimodular polytope.

A totally unimodular polytope is a polytope whose facets have integer equations, and with the property that any maximal, linearly independent collection of facets intersects in an integer point because their matrix has determinant $pm 1$. In particular the vertices are such intersections, so the vertices are all integral. This is a vast generalization of Birkhoff's theorem that comes from generalizing one of the proofs of Birkhoff's theorem.

Example: An alternating-sign matrix is equivalent to a square ice orientation of a square grid. The square ice orientations can be defined by a network flow, so you obtain an alternating-sign-matrix polytope. The generalized Birkhoff theorem in this case says that every vertex of the polytope is an alternating-sign matrix, in fact that every integer point of the $n$-dilated polytope is a sum of $n$ alternating-sign matrices.

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The other case that I encountered was the polytope of fractional perfect matchings of a non-bipartite set with $2n$ elements. By contrast, the Birkhoff polytope is the case of a bipartite set with $n$ elements of each type. By definition, it is the polytope of non-negative weights assigned to the edges of the complete graph on $2n$ vertices, such that the total weight at each vertex is 1. Strictly speaking, the Birkhoff theorem is false; not every vertex is a perfect matching. Instead, all of the vertices are combinations of matched pairs, and odd cycles with weight $frac12$.

At first glance this looks like bad news for the application of computing a perfect matching or the optimum perfect matching of a graph. Indeed, if instead you take the convex hull of the perfect matchings, the result is a polytope with exponentially many facets. However, a good algorithm exists anyway; there is a version of the simplex algorithm that only ever uses polynomially many of the facets.

nt.number theory - Order of the Tate-Shafarevich group

The first example of an abelian variety with nonsquare Sha was discovered in a computation by Michael Stoll in 1996. He emailed it to me and Ed Schaefer, because his calculation depended on a paper that Ed and I had written. At first none of us believed that it was what it was: instead we thought it must be due to either an error in Stoll's calculations or an error in the Poonen-Schaefer paper. Stoll and I worked together over the next few weeks to develop a theory that explained the phenomenon, and this led to the paper http://math.mit.edu/~poonen/papers/sha.ps - that paper contains a detailed answer to your question.

To summarize a few of the key points: If the abelian variety over a global field $k$ has a principal polarization coming from a $k$-rational divisor (as is the case for every elliptic curve), then the order of Sha is a square (if finite), because it carries an alternating pairing - this is what Tate proved, generalizing Cassels' result for elliptic curves. For principally polarized abelian varieties in general, the pairing satisfies the skew-symmetry condition $langle x,y rangle = - langle y,x rangle$ but not necessarily the stronger, alternating condition $langle x,x rangle=0$, so all one can say is that the order of Sha is either a square or twice a square (if finite). Stoll and I gave an explicit example of a genus 2 curve over $mathbf{Q}$ whose Jacobian had Sha isomorphic to $mathbf{Z}/2mathbf{Z}$ unconditionally (in particular, finiteness could be proved in this example).

If the polarization on the abelian variety is not a principal polarization, then the corresponding pairing need not be even skew-symmetric, so there is no reason to expect Sha to be even within a factor of $2$ of a square. And indeed, William Stein eventually found explicit examples and published them in the 2004 paper cited by Simon.

A final remark: Ironically, my result with Stoll quantifying the failure of Sha to be a square is used by Liu-Lorenzini-Raynaud to prove that the Brauer group $operatorname{Br}(X)$ of a surface over a finite field is a square (if finite)!

soft question - Theorem versus Proposition

Of course, this is a very subjective question, but I would tend to use "Theorem" only for a statement which has genuine content (whether my own, or one I am citing) and which I wouldn't expect the reader to be able to prove themselves fairly easily. Usually a paper shouldn't have many of these, probably no more than one per section.

"Proposition" I would use after having given a definition, when showing that some fairly straightforward (but not completely obvious) consequence holds; for instance showing that some linear subspace of functions is actually a subalgebra. This is probably close to how you said you use "claim", although I suppose the difference is that you can propose something somewhat out of the blue following a definition, while "claim" is usually directly related to some logical structure which is already moving forward, say to highlight a point midway through the proof of a theorem.

So I make the distinction that Proposition is something that the reader, if so inclined, could easily prove for themselves once they understand the definition. It highlights a result that could just as well have been stated in plain text, emphasizing that while it may be straightforward to prove, it is nevertheless worthy of note.

ct.category theory - Explicit description of a fibered category

This construction may not be the most natural (or general) one, but I find it reasonably enlightening.

Let $mathcal F$ and $mathcal C$ denote the categories with one object associated to $G$ and $H$, respectively. Notice that if $mathcal F'$ is any category equivalent to $mathcal F$, then in particular it admits a fully faithful functor to $mathcal F$. Since $mathcal F$ has only one object, with isomorphisms in bijection with $G$, this implies that every hom-set in $mathcal F'$ must be in bijection with $G$ as well. It's easy to check that every morphism in $mathcal F'$ must be an isomorphism, so this proves that $mathcal F'$ is a groupoid, with exactly $|G|$ isomorphisms between any two objects.

I claim that we can choose $mathcal F'$ to have objects indexed by $H$. To be explicit, let's say that the morphisms between any two objects $h_1, h_2$ are identified with $G$, and that the composition of $g_1: h_1 to h_2$ and $g_2: h_2 to h_3$ is $g_1 g_2: h_1 to h_3$. Then this admits a natural "projection" functor to $mathcal F$, by sending every object to the unique object $*$ of $mathcal F$ and sending each morphism to the morphism of the same name. We get a functor in the other direction by sending $*$ to the object labeled by the identity of $H$, and preserving names of morphisms. The composition $mathcal F to mathcal F' to mathcal F$ is literally the identity functor, and $mathcal F' to mathcal F to mathcal F'$ is easily seen to be naturally isomorphic to the identity functor via a base-preserving natural transformation. So $mathcal F'$ and $mathcal F$ are equivalent fibered categories over $mathcal C$.

Now let's construct a splitting of $mathcal F' to mathcal C$. Fix a representative $widetilde h in G$ for each element $h in H$. Consider the subcategory of $mathcal F'$ that includes all objects, but only the morphisms of the form $widetilde h_1 widetilde h_2^{-1}: h_1 to h_2$. (Note that this does contain identities and compositions.) For any given morphism $h$ in $mathcal C$ and any object $h_2 in mathcal F'$, our chosen subcategory contains a unique pullback $h_1 to h_2$ of $h$, namely the morphism $widetilde h_1 widetilde h_2^{-1}: h_1 to h_2$ with $h_1$ chosen so that $h_1 h_2^{-1} = h$.

To make this more concrete, let's look at the simplest possible non-split group extension: $mathbb Z/4mathbb Z twoheadrightarrow mathbb Z/2mathbb Z$. Here, the category $mathcal F'$ has two objects, with four morphisms between any pair, all of them isomorphisms. This category deserves to be equivalent to $mathcal F$: it has two objects, which both look exactly like the object of $mathcal F$ and are isomorphic to each other. Make $mathcal F'$ into a fibered category over $mathcal C$ by composing the "projection" functor to $mathcal F$ with the given functor $mathcal F to mathcal C$. Recall that we can't construct a splitting of the original fibered category $mathcal F to mathcal C$ precisely because we would need to choose a lift of the morphism $1 in mathbb Z/2mathbb Z$ to $mathbb Z/4mathbb Z$, and neither of the two choices gives something that respects composition. But in our new fibered category $mathcal F'$, we need to choose a lift of $1 in mathbb Z/2mathbb Z$ to some morphism between the two objects of $mathcal F'$, instead of an automorphism of one of the objects. So we don't need to worry about composing the lift with itself, and the problem is avoided.