parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/223215 | 3 |
>
> Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated
> subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must
> $A \cap B$ be finitely generated?
>
>
>
Recall that a limit group is a group whose existential theory (the set of true sentences in first order theory which use only... | https://mathoverflow.net/users/38889 | Do limit groups satisfy Howson's theorem? | The answer is 'yes'. A geometric proof, showing in fact that every finitely generated subgroup of a limit group is relatively quasiconvex, was given by [Dahmani](http://arxiv.org/abs/math/0203258).
| 6 | https://mathoverflow.net/users/1463 | 223223 | 104,536 |
https://mathoverflow.net/questions/223221 | 4 | Let $X$ be a smooth projective curve, and $E$ a vector bundle on $X$ such that there exist a bilinear perfect symmetric form $$E\otimes E\rightarrow \mathcal O\_X$$
When I see $E$ as a $GL\_r$ principal bundle, the existence of such a form is equivalent to the existence of a reduction of the structure group to the l... | https://mathoverflow.net/users/75343 | Vector bundles with symmetric perfect form | You get this by picking a frame $(e\_1,\dots,e\_r)$ of $E|\_U$ for a neighbourhood $U$ of each point $x$ such that $(e\_i,e\_j)=\delta\_{ij}$. As pointed out by t3suji, these neighbourhoods are étale in general, because they have to carry some square roots of regular functions. If you use such frames to trivialise your... | 3 | https://mathoverflow.net/users/70808 | 223225 | 104,537 |
https://mathoverflow.net/questions/223226 | 15 | I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that Zagier was the first to find the triangle but no hard reference as to where one can look up the ideas used.
Does anyon... | https://mathoverflow.net/users/82634 | Reference to a Don Zagier Result and the Congruent Number Problem | Don Zagier's own explanation is [here](http://people.mpim-bonn.mpg.de/zagier/files/mpim/89-23/fulltext.pdf), page 4+5 (in German, my translation).
>
> Consider the elliptic curve given by the equation $y^2=x(x+n)(x-n)$.
> If $P=(x,y)$ is an arbitrary nontrivial solution (meaning $y\neq 0$)
> of this equation, the... | 17 | https://mathoverflow.net/users/11260 | 223238 | 104,542 |
https://mathoverflow.net/questions/223206 | 4 | Start with a permutation $s\_0$ of the numbers
$(1,\ldots,n)$, e.g., for $n=10$,
$s\_0=(8,2,1,6,9,7,10,5,4,3)$.
Form $s\_1$ by using the numbers in $s\_0$ as indices into $s\_0$.
So $s\_1$ is composed of the $8$-th
element of $s\_0$, followed the $2$-nd element,
followed by the $1$-st element, followed
by the $6$-th el... | https://mathoverflow.net/users/6094 | Why does iterated indexing avoid cycles of length 5? | Let's look at your permutation $s\_0 = (8, 2, 1, 6, 9, 7, 10, 5, 4, 3)$. We can rewrite this in the cycle notation as (1, 8, 5, 9, 4, 6, 7, 10, 3, 1) (2) and so we can see that it has order 9 - that is, $s\_0^9 = 1$.
Now $s\_i = (s\_0)^{2^i}$, as observed by user78588. In particular $s\_6 = (s\_0)^{64} = (s\_0)^{7\t... | 6 | https://mathoverflow.net/users/143 | 223241 | 104,545 |
https://mathoverflow.net/questions/223180 | 9 | I'm given two matrices in $SL\_2(\mathbb{Z})$
$$
A = \left(\begin{array}{cc}
2 & 3\\
3 & 5
\end{array}\right), \ \
B = \left(\begin{array}{cc}
5 & 3\\
3 & 2
\end{array}\right).
$$
Then the group $\langle A, B \rangle$ is free of rank 2. Now my problem is to prove that $\langle A, B \rangle$ does not contain a... | https://mathoverflow.net/users/23754 | Integer matrix that does not belong to a free group of rank 2 | This can be proved using the [modular diagram](http://www.uncg.edu/mat/numbertheory/Gallery/images/quadratic-form-tree.png) in the upper half plane, and the dual tree $T$ of the modular diagram (see below for a discussion of the [picture in the link](http://www.uncg.edu/mat/numbertheory/Gallery/images/quadratic-form-tr... | 13 | https://mathoverflow.net/users/20787 | 223242 | 104,546 |
https://mathoverflow.net/questions/223129 | 14 | Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).
Define the *greedy* or *Demazure product* of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that w... | https://mathoverflow.net/users/391 | Coxeter exchanges in non-reduced words | (in contrary to what I thought first,) here is a proof that the "exchange condition" holds in the following sense.
It is based on the root configuration in <http://arxiv.org/abs/1111.3349> [1].
Let $(W,S)$ be a Coxeter system, $Q \in S^\*$ a word in $S$, $w \in W$, $P$ a subword of $Q$. We everywhere consider subword... | 4 | https://mathoverflow.net/users/21291 | 223243 | 104,547 |
https://mathoverflow.net/questions/222583 | 10 | There is a classical result commonly attributed to W. Sierpiński that reads as follows:
>
> **Theorem 1.** If $f: \Sigma \to \bf R$ is a non-atomic (\*) measure on a set $S$, then for every $X \in \Sigma$ and $a \in [0, f(X)]$ there exists $A \in \Sigma$ such that $A \subseteq X$ and $f(A) = a$.
>
>
>
In an a... | https://mathoverflow.net/users/16537 | A result of Sierpiński on non-atomic measures | I don't yet have a reference, but it seems the result might have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
>
> R. Sikorski, *Real Functions, Vol. 1*, PWN: Warsaw, 1958 (in Polish),
>
>
>
at least according to the histo... | 6 | https://mathoverflow.net/users/16537 | 223245 | 104,549 |
https://mathoverflow.net/questions/222578 | 6 | While preparing a lecture on dynamic programming principle in optimal stochastic control after the [book of Touzi](http://www.springer.com/us/book/9781461442851), I discovered a gap in the proof of DPP (page 28 of the book).
Here I simplify the relevant place to give a brief idea where the gap lies. The author defin... | https://mathoverflow.net/users/8146 | Countable subcover of half-open cylinders | Let $B\_r$ denote the closed ball of radius $r$ centered at $0$ in $\mathbb{R}^d$.
*Claim*: for any $r>0$ and $\tau <T$ there is a countable subfamily of $D$ that covers $[0,\tau] \times B\_r$.
To prove the claim, consider the set $I$ of all $t\in [-\infty, \tau]$ such that $(t, \tau] \times B\_r$ is covered by so... | 1 | https://mathoverflow.net/users/6101 | 223248 | 104,551 |
https://mathoverflow.net/questions/223153 | 2 | According to Mattila, *Geometry of sets and measures in Euclidean spaces*, p. 168, the **Fourier dimension** $\text{dim}\_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $0<s<\text{dim}\_F A$ there exists a non-zero Radon measure $\mu$ with spt $\mu\subset A$ and $|\hat\mu(x)|\le |x|... | https://mathoverflow.net/users/4600 | Lower bounds from Fourier dimension? | Fourier dimension doesn't directly say anything about lower bounds for the mass of balls. Fourier dimension is smaller or equal than Hausdorff dimension, and in order to give a bound of the form $\mu(B(x,r))\ge r^s$, the exponent $s$ needs to be large.
This suggests that one should seek a notion of dimension that giv... | 3 | https://mathoverflow.net/users/11009 | 223253 | 104,553 |
https://mathoverflow.net/questions/223250 | 4 | Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D\_I$ of the algebra $\mathcal{A}/\mathcal{I}$.
The question is given $D'\in Der(\mathcal{A}/\mathcal{I})$ there exist $D\i... | https://mathoverflow.net/users/82637 | Covering derivations of a quotient algebra | Let $\mathcal{A}$ be the two-dimensional Lie algebra over a field $k$ with basis $\{x,y\}$ and relation $[x,y] = y$. Let $\mathcal{I}$ be the ideal of $\mathcal{A}$ spanned by $y$. Let $D' : \mathcal{A} / \mathcal{I} \to \mathcal{A}/\mathcal{I}$ be the $k$-linear map that sends the image $\overline{x}$ of $x$ in $\math... | 4 | https://mathoverflow.net/users/6827 | 223254 | 104,554 |
https://mathoverflow.net/questions/222478 | 14 | I was thinking about [this question](https://math.stackexchange.com/q/1508821/19661) asked at Math.SE, when I came up with the following conjecture.
For every $q\in\mathbb Q$ consider a sequence $s\_n^{(q)}$ (terms within the sequence are indexed by $n\in\mathbb N$):
$$s\_n^{(q)}=\left\lfloor\frac{n!\cdot q}e\right\r... | https://mathoverflow.net/users/9550 | A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$ | Write
$$
\frac{n!\,e^{-1}}m=x\_n+\frac{(-1)^{n+m+1}y\_n}m+\frac{(-1)^{n+1}\delta\_n}m
\qquad\text{and}\qquad
\frac{!n}m=x\_n+\frac{(-1)^{n+m+1}y\_n}m,
$$
where
$$
x\_n=\frac{n!}{m\cdot(n-m)!}\cdot(n-m)!\sum\_{k=0}^{n-m}\frac{(-1)^k}{k!}\in\mathbb Z,
\\
y\_n=n!\sum\_{k=n-m+1}^n\frac{(-1)^{k-(n-m+1)}}{k!}\in\mathbb Z,
$$... | 7 | https://mathoverflow.net/users/4953 | 223265 | 104,557 |
https://mathoverflow.net/questions/223228 | 4 | Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ such that $f\_\*\mathcal{L}$ is isomorphic to the trivial sheaf $\mathcal{O}\_Y$? If so, is there any literature on thi... | https://mathoverflow.net/users/58203 | A moduli problem inspired by Stein factorization | I don't know a reference but I think what I said in the comment is more or less obvious. Let me prove it though in detail. Again, I can't imagine one can make this into any reasonable moduli space.
**Setting:** Suppose that $\pi : X \to Y$ is a proper birational map with connected fibers with $X$ nonsingular and $Y$... | 7 | https://mathoverflow.net/users/3521 | 223269 | 104,558 |
https://mathoverflow.net/questions/223155 | 4 | As we know the Hessian matrix is symmetric in a finite-dimensional environment. What about the Hessian operator $D^2F$ for a functional $F:H\rightarrow \mathbb{R}$, where $H$ is a Hilbert space and $D$ is the Fréchet derivative?
More specifically, $D^2F(p)$ ($p$ is a critical point of $F$) plays an important role in ... | https://mathoverflow.net/users/82599 | Is Hessian operator self-adjoint on infinite dimensional environment? | $d^2F(p)$ is symmetric as a bilinear form. See 5.11 of [here](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf), for example.
| 4 | https://mathoverflow.net/users/26935 | 223274 | 104,560 |
https://mathoverflow.net/questions/223259 | 6 | Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$.
For any $q\_1, q\_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ such that $q\_2 = n \cdot q\_1$, we can form a surjective homomorphism $\Bbb Q/q\_2\Bbb Z \to \Bbb Q/q\_1\Bbb Z$.
Th... | https://mathoverflow.net/users/24611 | Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles? | For 1 you can indeed use duality, but you can also give an explicit isomorphism:
First for each rational number $q$ it is easy to see that $\mathbb{A}^{f}/(q \hat{\mathbb{Z}}) \simeq \mathbb{Q}/(q\mathbb{Z})$: indeed, $(q \hat{\mathbb{Z}})$ is open and $\mathbb{Q}$ is dense in the finite adeles, so the image from $\m... | 7 | https://mathoverflow.net/users/22131 | 223282 | 104,561 |
https://mathoverflow.net/questions/223273 | 6 | I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S\_g$ must itself already be homemorphic to $S\_g \times \mathbb R$. There seems to be a very deep theorem behind it, however, I couldn't find any reference which would bring this u... | https://mathoverflow.net/users/78554 | 3-manifolds homotopy equivalent to a surface | The answer (by studiosus) to [this question on MSE](https://math.stackexchange.com/questions/734107/a-3-manifold-with-fundamental-group-isomorphic-to-a-surface-group) gives references AND a counterexample!
| 2 | https://mathoverflow.net/users/11142 | 223293 | 104,565 |
https://mathoverflow.net/questions/223296 | 5 | I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
1. Why did people initially consider such units in $\mathbb{Q}(\zeta\_p)$ specially?
There are many interesting results we can get with the notion of cyclotomic units, for example, comparison between the... | https://mathoverflow.net/users/44005 | Motivation for cyclotomic units | I'm quite sure cyclotomic units were first introduced by Kummer in his 1847 paper, where he proved his very famous partial solution to Fermat's Last Theorem:
>
> **Theorem** (Kummer) If $p$ is an odd prime that does not divide the class number of the field $\mathbb{Q}(\zeta\_p)$ and $(xyz,p)=1$, then
>
>
> $$x^p+... | 7 | https://mathoverflow.net/users/43108 | 223298 | 104,567 |
https://mathoverflow.net/questions/223277 | 29 | In 1986, Don Zagier generalized Euler's theorem ($\zeta\_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta\_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum\_v c\_v A(x\_{v,1})...A(x\_{v,s})$$
where
$$A(x)=\int^x\_0 \frac{1}{1+t^2}\log \frac{4}{1+t^2}dt$$
In this sense, the result is conjectured... | https://mathoverflow.net/users/43108 | Intuition for Zagier's theorem for $\zeta_K(2)$ | As ThiKu mentions, the connection between $\zeta\_K(2)$ and hyperbolic manifolds is that the volume formula for arithmetic hyperbolic manifolds is given by an explicit formula involving $\zeta\_K(2)$. This formula is due to **[Borel](https://eudml.org/doc/83853)** in the case of arithmetic manifolds arising from quater... | 25 | https://mathoverflow.net/users/nan | 223299 | 104,568 |
https://mathoverflow.net/questions/220849 | 2 | Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}\_n(\mathbb{F}\_p)$ acts on the polynomial ring $\mathbb{F}\_p[x\_1,\ldots,x\_n]$ via $A\cdot x\_i=\sum\_j a\_{ji}x\_j$ for all $A=(a\_{ij})\in H$, and $A\cdot f(x\_1,\ldots,x\_n):=f(A\cdot x\_1,\ldots,A\cdot x\_n)$.
>
>
> >
> > H... | https://mathoverflow.net/users/32261 | Invariant polynomials under the action of $H\le\operatorname{GL}_n(\mathbb{F}_p)$ | Let me add to Chris' answer that there is indeed a degree bound: $\mathbb{F}[x\_1,\ldots,x\_n]^H$ is generated in degree at most $n (|H| - 1)$. The bound was proved by Peter Symonds in: *On the Castelnuovo-Mumford regularity of rings of polynomial
invariants*, Ann. of Math. (2) 174 (2011), 499–517.
And yes, a general... | 4 | https://mathoverflow.net/users/82616 | 223301 | 104,570 |
https://mathoverflow.net/questions/223292 | 0 | Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$.
Question A: Is $GL\_1(C(Y))\stackrel{\pi}{\longrightarrow} K\_1(C(Y))$ surjective?
Question B: If $r:C(Y)\rightarrow C(Y\setminus X)$ denotes the restricti... | https://mathoverflow.net/users/82659 | Nonstable $K$-theory question | Regarding Question A, the $3$-torus provides a counterexamle: its complex $K\_1$-group has rank $4$ but its cohomotopy has only rank $3$. (I have learnt about this example from the book of Rordam, Larsen and Laustsen.)
The diagram in Question B clearly commutes because $K\_1(r)$ just acts by picking some representati... | 2 | https://mathoverflow.net/users/1291 | 223308 | 104,573 |
https://mathoverflow.net/questions/223297 | 3 | In what follows "homology" will mean group homology, i.e. $H\_\*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.
It is well-known how to compute the homology of abelian groups, for example for a torsion-free abelian group A one has $H\_n(A)=A\wedge\_{\mathbf Z}A\wedge\_{\mathbf Z}\ldots\wedge\_{... | https://mathoverflow.net/users/39082 | Homology of solvable Lie groups made discrete | The example that you mention is the semidirect product for the multiplicative group $k^\times$ acting nontrivially on the additive group $k^+$. Its homology coincides with the homology of $k^\times$. To see this you can a Hochschild-Lyndon-Serre spectral sequence. The point is that $H\_i(Bk^\times;H\_j(Bk^+))=0$ when $... | 5 | https://mathoverflow.net/users/6666 | 223320 | 104,578 |
https://mathoverflow.net/questions/223202 | 9 | This is a general question that I have. Let $X$ be a projective variety over an algebraically closed field $k$. Let $L$ be an ample line bundle over $X$. Let $F$ be a vector bundle on $X$. We say that $F$ is $L$-semistable if for any coherent subsheaf $0\neq E\subset F$ of strictly lesser rank, $\mu\_L(E)\leq\mu\_L(F)$... | https://mathoverflow.net/users/70211 | Why do we need ampleness in the definition of stability/semistability | You are right, the condition that $L$ be ample can be weakened. In fact, on an $n$-dimensional normal projective variety $X$ one can measure (semi-)stability with respect to an arbitrary *movable* curve class $\alpha \in N\_1(X)\_{\mathbb R}$. A numerical curve class $\alpha$ is said to be movable if $D \cdot \alpha \g... | 7 | https://mathoverflow.net/users/44860 | 223329 | 104,582 |
https://mathoverflow.net/questions/223336 | 5 | I am looking at a polynomial of the entries of a matrix, and this polynomial is invariant under permutation of the rows or columns of the matrix. Is there a similar characterization as in the case of symmetric polynomials of this family? I.e. some simple bases.
Any references welcome!
| https://mathoverflow.net/users/39359 | Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions? | You have touched a vast subject called invariant theory. The direct product $G=S\_n\times S\_n$ of two copies of the symmetric group $S\_n$ on $n$ letters naturally acts on polynomials in variables $X\_{ij}$, for $1\leq i,j\leq n$:
if $(\gamma,\mu)\in G$ then $(\gamma,\mu)$ sends $X\_{ij}$ to $X\_{\gamma(i),\mu(j)}$. A... | 6 | https://mathoverflow.net/users/11100 | 223343 | 104,583 |
https://mathoverflow.net/questions/223287 | 8 | In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One then extends this to arbitrary opens by taking the limit over all the rational subsets inside the given open. However, in ... | https://mathoverflow.net/users/76748 | why don't (can't?) we sheafify the structure presheaf of an adic space | There is actually a sheafification process used in classical construction of rigid geometry, which is used to pass from the weak G-topology to the strong one (see section 9.2.2 of Bosch, Güntzer, Remmert's Non-Archimedean Analysis, for instance).
As to why people don't use it in the case of non-sheafy spaces, I woul... | 2 | https://mathoverflow.net/users/4069 | 223346 | 104,584 |
https://mathoverflow.net/questions/223344 | 12 | In this paper: <http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf>
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same result that did not rely on the Riemann mapping theorem.
Start with a simply connected domain, fill it with coins mak... | https://mathoverflow.net/users/16959 | Is there a proof of the uniformization theorem using circle packing? | Firstly, the Rodin-Sullivan argument can not, in principle, be used to give a proof of uniformization, since it uses uniformization of domains of infinite type (due to Marden, if I recall) as an ingredient.
Secondly, there are many circle-packing approaches to uniformization, the first of them given by G. Leibon, usi... | 6 | https://mathoverflow.net/users/11142 | 223347 | 104,585 |
https://mathoverflow.net/questions/223323 | 9 | **Question:** Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?
Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in $C$ there is a unique geodesic contained in $C$. (For example an open hemisphere is convex inside the sphere)
I hav... | https://mathoverflow.net/users/51663 | Is every open convex subset of a Riemannian manifold necessarily contractible? | Fix $q \in C$. The set $D\subseteq T\_q M$ of all tangent vectors $v$ such that $[0,1]\ni t \mapsto \exp\_q(tv)$ is the unique geodesic in $C$ connecting $q$ with $p$ is open and star shaped, hence contractible.
Then $\exp\_q: D \to C$ is a continuous map between manifods with the same dimension. It is also one-to-on... | 5 | https://mathoverflow.net/users/13915 | 223369 | 104,590 |
https://mathoverflow.net/questions/162657 | 16 | **Question:** Does every $\omega\_1$-Aronszajn tree contains a Suslin sub-tree or a special Aronszajn sub-tree?
Recall that Suslin trees are $\omega\_1$-trees (trees of height $\omega\_1$, and countable width) that have only countable anti-chains. Special Aronszajn trees are $\omega\_1$-trees that can be represented ... | https://mathoverflow.net/users/41953 | Does every Aronszajn tree has a Suslin or a Special subtree? | It is consistent that the answer is no. The following is proved in Beaodouin's thesis ``[On uncountable trees and linear orders](http://libarchive.dartmouth.edu/cdm/ref/collection/dcdis/id/149265)'', as Theorem 1.10:
>
> **Theorem.** Assume $\kappa^{<\kappa}=\kappa$ and $\Diamond(E)$ holds, where
> $E \subseteq \{... | 8 | https://mathoverflow.net/users/11115 | 223376 | 104,592 |
https://mathoverflow.net/questions/223356 | 5 | I've been working out how the internal language of a category **C** extends to taking the category itself as a type.
The most obvious way to interpret $X : \mathbf{C}$ is, of course, that $X$ is an object of **C**. We could ask for generalized elements too; e.g. we could consider functions $S \to \mathbf{Ob}(\mathbf{... | https://mathoverflow.net/users/nan | What is the right notion of generalized element of a category? | **Yes, it works well to take “functors into *C*” as the notion of “generalised element of *C*”.**
For groupoids, the Hofmann–Streicher model of type theory gives a strong precise statement of this: closed (i.e. non-dependent) types are interpreted exactly as groupoids **G**; closed elements of such a type are interpr... | 6 | https://mathoverflow.net/users/2273 | 223378 | 104,593 |
https://mathoverflow.net/questions/223189 | 6 | For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}\_K(\mathfrak{m}/\mathfrak{m}^2) - \operatorname{dim}(R).$$ Here are some questions:
1. If $P \subseteq R$ is prime ideal, is it always true that $\operato... | https://mathoverflow.net/users/82616 | Is the embedding dimension minus the dimension upper semicontinuous? | Edit. Finally the proof was not so long, so I include it complete:
Question 3. Embedding codimension (sometimes simply codimension).
Question 1. I don't have access here to "Lech, Inequalities related to certain couples of local rings, Acta math. 112 (1964), 69-89", but maybe that paper will answer it better, I can... | 2 | https://mathoverflow.net/users/36672 | 223390 | 104,595 |
https://mathoverflow.net/questions/223391 | 0 | Is it possible to construct a set of numbers of arbitrary size such that any calculation involving addition and subtraction, on any combination of those numbers, produces a unique result?
For example, the set of numbers 1,2,3,4,5 would not satisfy this condition because:
* 1 + 2 = 3
* 1 + 5 = 2 + 4 = 6
* 1 + 2 = 5 ... | https://mathoverflow.net/users/82704 | Set of number with unique sums of elements | Taking different powers of 3, i.e.
1,3,9,27,81,... should do the job.
| 3 | https://mathoverflow.net/users/78588 | 223392 | 104,596 |
https://mathoverflow.net/questions/223380 | 1 | Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism **defined over $\overline{\mathbb{Q}}$**
$$f: A\_{\overline{\mathbb{Q}}}\rightarrow \tilde{A}\_{\overline{\mathbb{Q}}}$$
can be twisted t... | https://mathoverflow.net/users/70751 | Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension? | In good cases, the ''isomorphism functor'' $\mathrm{Isom}(X,Y)$ is representable by a scheme. Hence, you are asking that if this scheme is non-empty (i.e. contains a geometric point), whether it contains a solvable point. So your problem is more-or-less equivalent to the usual open problem that every variety contains a... | 6 | https://mathoverflow.net/users/5101 | 223409 | 104,602 |
https://mathoverflow.net/questions/223305 | 2 | Let $G=(V,E)$ be a graph and $V=A\cup B$ satisfying
$(1)A\cap B=\emptyset;$
$(2)|N\_G(v)\cap B|\geq |N\_G(v)\cap A|,\forall v\in A$ and $|N\_G(v)\cap A|\geq |N\_G(v)\cap B|,\forall v\in B$.
Let $S=\{(u,v)|u\in A,v\in B,N\_G(u)\cap N\_G(v)\neq \emptyset\}$ and $G[A],G[B]$ be the induced subgraphs of $G$ by $A$ an... | https://mathoverflow.net/users/58096 | A question about a specific partition of a graph | Denote $d\_A(v)=|N\_G(v)\cap A|$, $d\_B(v)=|N\_G(v)\cap B|$. Set $S\_A=\{(a,b)\in A\times B\colon N\_G(a)\cap N\_G(b)\cap A\neq\varnothing\}$ and $S\_B=\{(a,b)\in A\times B\colon N\_G(a)\cap N\_G(b)\cap B\neq\varnothing\}$. Clearly, $S\_A,S\_B\subseteq S$. Now we claim that $|S\_A|\geq 2|E(G[A])|$ and, similarly, $|S\_... | 3 | https://mathoverflow.net/users/17581 | 223414 | 104,604 |
https://mathoverflow.net/questions/223247 | 7 | This is a followup to [here](https://mathoverflow.net/questions/220827/lemma-1-from-beilinsons-coherent-sheaves-on-mathbbpn-and-problems-of-lin/).
Consider Lemma 2 from Beilinson's paper "[Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra](http://www.ams.org/mathscinet-getitem?mr=509388)", as follows.... | https://mathoverflow.net/users/nan | Lemma 2 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition? | $\newcommand\Ocal{\mathcal{O}}\newcommand\Hom{\mathop{\mathrm{Hom}}}\newcommand\Ext{\mathop{\mathrm{Ext}}\nolimits}$I think the following is a rather clumsy way of proving it. It is an induction on two variables, but probably there are better ways to set up the induction. The idea is to prove it for
1. $i$ between 0 ... | 5 | https://mathoverflow.net/users/6263 | 223418 | 104,606 |
https://mathoverflow.net/questions/223408 | 4 | I have already posted [this question](https://math.stackexchange.com/questions/1519357/invariant-subspace-vs-reducing-subspace-in-some-special-hilbert-space) on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question.
Recently I was reading a book "Operator Function and syste... | https://mathoverflow.net/users/82713 | Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure | I guess it depends on what counts as simple or elementary, but I think the standard argument goes like this: using the F. and M. Riesz Theorem, one can show that since $\mu$ is singular to Lebesgue measure, there is a sequence of analytic polynomials $p\_n$ such that $p\_n(z)\to \overline{z}$ weak-\* in $L^\infty(\mu)$... | 6 | https://mathoverflow.net/users/13360 | 223419 | 104,607 |
https://mathoverflow.net/questions/223430 | 1 | Been going through Alan Baker's A Comprehensive Course in Number Theory. Very interesting book, although the way proofs are presented sometimes throws me off a little.
I usually read through a chapter multiple times and then try and solve a few exercises (I can't say I'm proficient or smart enough to solve all of th... | https://mathoverflow.net/users/82719 | Relationship between quadratic residues modulo a prime and quadratic residues modulo a prime power | I think a simple induction suffices. Suppose that $x^2\equiv a\pmod{p^k}$ with integer $x$ and $k$; that is, $x^2=a+tp^k$ where $t$ is also an integer. Then for any integer $n$, we have $(x+np^k)^2\equiv a+(t+2nx)p^k\pmod{p^{k+1}}$. Choosing $n$ to satisfy the linear congruence $t+2nx\equiv 0\pmod p$, we get $(x+np^k)^... | 12 | https://mathoverflow.net/users/9924 | 223434 | 104,611 |
https://mathoverflow.net/questions/223294 | 6 |
>
> Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.
>
>
>
(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric billinear form of signature $(3,2)$, and have determinant $1$.)
>
> $G$ is a group of real rank $2$, so c... | https://mathoverflow.net/users/38889 | Are the integer matrices in SO(3,2) "boundedly generated"? | $\mathrm{SO}(3,2)\_{\mathbb{Z}}$ is an orthogonal group of $\mathbb{Q}$-rank 2, so it is boundedly generated. This was proved by Tavgen [Math. USSR. Izvestiya 36 (1991) 101-127; [MR1044049](http://www.ams.org/mathscinet-getitem?mr=MR1044049)] for the quasi-split case, which includes $\mathrm{SO}(3,2)\_{\mathbb{Z}}$, bu... | 6 | https://mathoverflow.net/users/68305 | 223435 | 104,612 |
https://mathoverflow.net/questions/223314 | 1 | Let $c\_{0}\widehat{\otimes}\_{\pi} c\_{0}$ be the projective tensor product of $c\_{0}$ and $c\_{0}$. Let $(e\_{n})\_{n}$ be the unit vector basis of $c\_{0}$. For each $n$, define $z\_{n}=e\_{n}\otimes\sum\_{j=1}^{n}e\_{j}$. Then the basic sequence $(z\_{n})\_{n}$ is sub-symmetric. My question: Is each normalized blo... | https://mathoverflow.net/users/41619 | On the normalized block basic sequences in $c_{0}\widehat{\otimes}_{\pi} c_{0}$ | No. Consider $z\_{2n+1} - z\_{2n}$.
| 1 | https://mathoverflow.net/users/2554 | 223439 | 104,615 |
https://mathoverflow.net/questions/223342 | 6 | Suppose that $f,g\_1,g\_2,\dots$ are functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f'=f\,g\_1$ and $g'\_j=g\_j^2-g\_j g\_{j+1}$. Here and in what follows, $j$ is any natural number. Then, by induction, $f^{(j)}=f\, P\_j(g\_1,\dots,g\_j)$, where
$$P\_j(u\_1,\dots,u\_j)=\sum\_{k\_1=0}^j\cdots\sum\_{k\_j=0}^j c\... | https://mathoverflow.net/users/36721 | A property of the derivatives of a function | We have, writing for short $P\_n=P\_n(g\_1,\dots,g\_{n })$,
$$\big(f^{(n)}\big)' =\big(fP\_{n }\big)'=f' P\_{n }+\sum\_{i=1}^n f\partial\_iP\_ng\_i'=fg\_1 P\_{n }+\sum\_{i=1}^n f\partial\_iP\_n(g\_i^2-g\_ig\_{i+1})$$
Comparing with $f^{(n+1)}=fP\_{n+1}$ we obtain a linear recursion for the sequence of polynomials $P\... | 9 | https://mathoverflow.net/users/6101 | 223441 | 104,616 |
https://mathoverflow.net/questions/223442 | 2 | For the theory of classical modular forms, the space of new forms $S\_k^{new}(\Gamma(N))$ has a basis of Hecke eigenforms $\{ f\_i = \sum a\_n q^n : a\_1=1, a\_n \in \bar{\mathbb{Q}}\}$
Given $k$ and $N$ (I'll take answers restricted to $\Gamma\_0(N)$ or $\Gamma\_1(N)$)
1) can we say anything about $\{dim\_{\mathbf... | https://mathoverflow.net/users/47195 | How do the Dim($K_f / \mathbf{Q}$) vary for all f in a given $S_k(\Gamma(N))$? | The answer to your last question is no. In general, if $K\_f/\mathbb Q$ has degree $d$, then there is an associated factor of $J\_0(N)$ defined over $\mathbb Q$ of dimension $d$. This is all in Shimura's *Arithmetic Theory of Automorphic Forms*, I think.
For your question (2), the curves $X\_0(N)$ and $X\_1(N)$ are d... | 7 | https://mathoverflow.net/users/11926 | 223445 | 104,617 |
https://mathoverflow.net/questions/223440 | 4 | Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}\_p}$ but is not algebraically closed?
| https://mathoverflow.net/users/45397 | Examples of perfect pseudo algebraically closed fields in positive characteristic | Yes, every ultraproduct $F$ of fields of the form $\mathbb{F}\_{p^{n!}}$ has this property. It is pseudo finite and hence perfect and pseudo algebraically closed. Moreover, since every polynomial with coefficients in $\mathbb{Z}$ splits in $\mathbb{F}\_{p^{n!}}$ for sufficiently large $n$, the field $F$ contains an alg... | 6 | https://mathoverflow.net/users/54365 | 223450 | 104,619 |
https://mathoverflow.net/questions/223449 | 2 | I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).
The resolutions mentioned in 4th and 5th paragraphs of [this question](https://mathoverflow.net/questions/136590/where-should-i-search-f... | https://mathoverflow.net/users/82725 | dg-resolution of the polynomial algebra | I'll give more of a reference to a general theory rather than the specifics of an ad hoc argument for why these are resolutions. There are many places where versions of this are written up. I am personally partial to the account in Loday--Vallette's Algebraic Operads (chapter 3), but this example is in Priddy's origina... | 1 | https://mathoverflow.net/users/3075 | 223462 | 104,623 |
https://mathoverflow.net/questions/223465 | 2 | Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X\_t)] = \mathbb E[f(\hat X\_t)], \ \forall t>0$$
I wonder if it is safe to generalize the above identity to a hitting time. More precisely, we d... | https://mathoverflow.net/users/5656 | Is it safe to work on a Cadlag modification of a Feller process? | I think the following example says no.
Consider the state space $\{0,1\}$. Let $U$ be a uniform random variable on $[0,1]$ and let $X\_t = 1$ if $t=U$ and $X\_t = 0$ otherwise. Note that $X\_t$ is a.s. not cadlag, but for each $t$ we have $X\_t = 0$ a.s., so $X\_t$ is a modification of the cadlag process $\hat{X}\_t$... | 2 | https://mathoverflow.net/users/4832 | 223471 | 104,625 |
https://mathoverflow.net/questions/223453 | 3 | Let $\mathcal{X}$ be a *canonical* stack (**edit:** I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. just one reduced component) on $X$, and $\mathcal{D}=\pi^{-1}(D)$ the pulled back divisor on $\mathcal{X}$, which wi... | https://mathoverflow.net/users/4721 | Question about canonical DM stacks | I'm writing an answer that expands on my last comment:
I assume that your $D$ on $X$ is an effective Cartier divisor, that I see as an invertible sheaf $\mathcal{O}(D)$ on $X$ with a global section $1\_D$.
Then its pullback $\mathcal{D}$ as a divisor will be given by $(\pi^\*\mathcal{O}(D),\pi^\*1\_D)$. If the stac... | 3 | https://mathoverflow.net/users/5516 | 223475 | 104,627 |
https://mathoverflow.net/questions/223333 | 6 | As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $\*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $\*$-sub-algebra is an FRT-algebra, which is to say, roughly, that they can be constructed from an R-matrix in the Yang--Baxter sense. (This goes back to the roo... | https://mathoverflow.net/users/81477 | Compact Quantum Groups and FRT-Algebras | Already for ${\rm SU}\_q(n)$ you have to remember that there is an extra relation about the quantum determinant being $1$. Anyhow, the correspondence you mention comes the fact that intertwiners of finite dimensional representations of ${\rm SU}\_q(n)$ can be constructed from eigenspace decomposition of the braiding ac... | 2 | https://mathoverflow.net/users/9942 | 223478 | 104,629 |
https://mathoverflow.net/questions/223470 | 13 | Ever since my exposure to [this integral](https://math.stackexchange.com/questions/958624/prove-that-int-01-sin-pi-xxx1-x1-x-dx-frac-pi-e24?lq=1) involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral
$$\int\_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, \quad m = 0, \,... | https://mathoverflow.net/users/70508 | $\pi e$ and an unfamiliar polynomial | Ahhh, I see where something went "off":
Consider
$$f(z) = \exp{\left( i \pi q z+ (m+qz)\log(z) + q(1-z)\log(1-z)\right)} $$
where $m$ is a non-negative integer and $q$ is a integer. Using the exact same branch cuts and contour (just about every single piece of information stays true, with the exception of some a... | 5 | https://mathoverflow.net/users/70508 | 223480 | 104,630 |
https://mathoverflow.net/questions/223467 | 2 | I'm looking for references to constructions and treatments of Hida Families/Eigenvarieties for ordinary Siegel modular forms (In particular: genus 2).
So far I've been reading Richard Taylor's thesis and would like to find more.
Thanks in advance.
| https://mathoverflow.net/users/47195 | Families of ordinary Siegel Modular Forms | This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results -- you can try to interpolate any or all of { Hecke eigenvalues, Fourier coefficients, L-values, Galois representations }, for forms satisfying various different flavours of finite-slope con... | 2 | https://mathoverflow.net/users/2481 | 223483 | 104,631 |
https://mathoverflow.net/questions/223489 | 1 | Let say we have an arbitrary polynomial over the reals, and we do not know whether it is separable or not. Is there some algorithm to find the number of roots it has in the complex number?
| https://mathoverflow.net/users/82741 | Is there a way to find out how many distinct roots a polynomial has? | If $f(z) = \prod (z-z\_i)^{n\_i}$, with $z\_i$ distinct, then $GCD(f(z), f'(z)) = \prod (z-z\_i)^{n\_i-1}$, so the number of distinct complex roots is $\deg f - \deg GCD(f, f')$. If $f$ has rational coefficients, or in some other sense can be computed with exactly, this is a practical method; polynomial GCD can be comp... | 7 | https://mathoverflow.net/users/297 | 223494 | 104,636 |
https://mathoverflow.net/questions/223417 | 2 | When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature.
In other words, do they implicitly admit that they use the Legendre orthogonal polynomials by default?
| https://mathoverflow.net/users/82192 | Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature？ | It is standard to use the term Gaussian Quadrature to refer broadly to any approximation
$$\int\_a^b \omega (w)f(x)dx=\sum\_{i=1}^n w\_if(x\_i)$$
that is optimanl in the sense of Gauss's fundamental theorem of quadrature, *regardless of the associated polynomial*.
In particular, unless the context makes it someho... | 0 | https://mathoverflow.net/users/43108 | 223496 | 104,637 |
https://mathoverflow.net/questions/223479 | 5 | Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)?
If yes, a reference would be helpful.
**EDIT:** If the question is too general, may be one can classify somehow non-negatively curved compact 2-dimensional Alexandrov spaces,... | https://mathoverflow.net/users/16183 | Classification of 2-dimensional Alexandrov spaces | Applying doubling theorem, we can get rid of boundary.
Then pass to the universal cover.
The obtained space is isometric to convex surface in the model space with curvature $\kappa$ and any motion of your original space can be extended as a motion of this surface.
So your space is a quotient of convex surface by a di... | 5 | https://mathoverflow.net/users/1441 | 223503 | 104,640 |
https://mathoverflow.net/questions/223524 | 2 | I want to read about graded rings and modules. First, I saw *Bruns-Herzog*. But it was difficult for a beginner. Then I saw notes of *Tom Marley* (see [Tom Marley's Homepage](http://www.math.unl.edu/~tmarley1/): Graded rings and modules). It is good. but it has a lot of exercises.
What other references do you know a... | https://mathoverflow.net/users/82759 | graded rings and modules | The book `*"Graded Syzygies"* by *"Irena Peeva"*`.
It starts with a grading on the polynomial ring. So it is good for a beginner. Preface of the book is expressive :
The main goal of the book is to inspire the readers and develop
their intuition about syzygies and Hilbert functions. Research on
free resolutions a... | 1 | https://mathoverflow.net/users/47763 | 223527 | 104,648 |
https://mathoverflow.net/questions/223431 | 3 | In Lurie's book Higher Algebra, he makes the following definition:
**Definition 3.1.2.2:** Let $M^\otimes\to N(Fin\_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another $\infty$-operad $B^\otimes$, let $q:C^\otimes\to O^\otimes$ be a fibration of $\infty$-operads and let $\overline{... | https://mathoverflow.net/users/11546 | Definition of Left Operadic Kan Extension for $\infty$-operads | Aaron's chat room remark is right: when restricted to $(M^\otimes)\_{/b}$ it is the canonical projection, and the cone point is sent to $b$.
Notice that for each object in the slice $f : x \to b$, there is a unique morphism from $f$ to the cone point in $(M^\otimes)\_{/b}^\vartriangleright$; this unique morphism is s... | 3 | https://mathoverflow.net/users/644 | 223528 | 104,649 |
https://mathoverflow.net/questions/223522 | 2 | Let $A$ be a local ring with maximal ideal $m$. Suppose that there exists some positive integer $k$ such that $m^k = m^{k+1}$.
Is necessarily $m^k = 0$ ?
If $m$ is finitely generated, this follows from Nakayama's lemma.
| https://mathoverflow.net/users/38889 | Can powers of a maximal ideal stabilize without vanishing? | The first thing that one might try seems to be a counter-example to that:
$R=k[x\_1,x\_2,\dots]$
$\mathfrak m={(x\_1,x\_2,\dots)}, I=(x\_1-x\_2^2,x\_2-x\_3^2,x\_3-x\_4^2,\dots)\subset R$.
$A=(R/I)\_{\mathfrak m}, m=\mathfrak m\_{\mathfrak m}$.
| 4 | https://mathoverflow.net/users/10076 | 223531 | 104,652 |
https://mathoverflow.net/questions/223517 | 9 | Let $S=\{\delta<\omega\_2:\text{cf}(\delta)=\omega\}$. A well-known theorem of Shelah tells us that we can find $\langle C\_\delta:\delta\in S\rangle$ such that
for every club $C\subseteq\omega\_2$ there are stationarily many $\delta\in S$ with $C\_\delta\subseteq C$, that is, $\langle C\_\delta:\delta\in S\rangle$ is ... | https://mathoverflow.net/users/18128 | Club-guessing at $\omega_2$ | YES for any given $\sigma<\omega\_1$. And you can even have the clubs (somewhat) cohere. I worked out the details in here: <http://blog.assafrinot.com/?p=2133>
| 11 | https://mathoverflow.net/users/20033 | 223533 | 104,653 |
https://mathoverflow.net/questions/223543 | 2 | What is the equation of motion for the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G \partial\_\mu G^{-1})$$where $G$ is an $N \times N$ unitary matrix? Could anyone supply a reference to its computation?
| https://mathoverflow.net/users/82765 | Equation of motion for the Lagrangian $\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$, $G$ is unitary $N \times N$ matrix? | The variation of the Lagrangian is fairly straight forward, keeping in mind the identity $\delta G^{-1} = - G^{-1} (\delta G) G^{-1}$ and the fact that $\mathrm{Tr}(AB)$ is a non-degenerate bilinear form.
\begin{align\*}
\delta\mathcal{L}
&= \mathrm{Tr}(\partial\_\mu \delta G \partial^\mu G^{-1})
- \mathrm{Tr}(\part... | 5 | https://mathoverflow.net/users/2622 | 223545 | 104,657 |
https://mathoverflow.net/questions/156408 | 28 | Given compact Lie groups $G$ and $\Pi$, there is a notion of "$G$-equivariant principal $\Pi$-bundle", and a corresponding notion of classifying space, often denoted $B\_G\Pi$, so that $G$-equivariant maps $X\to B\_G\Pi$ (up to equivariant homotopy) correspond to $G$-equivariant $\Pi$-bundles on $X$ (up to equivalence)... | https://mathoverflow.net/users/437 | Equivariant classifying spaces from classifying spaces | **Added Aug 2016:** I've written this up, available at <https://arxiv.org/abs/1608.02999>
$\def\Hom{\mathrm{Hom}} \def\Map{\mathrm{Map}} \def\ad{\mathrm{ad}}$
I think this is true. I'll sketch a possible proof here. I haven't
carefully checked everything, and there are things that need checking. Feel free to do tha... | 13 | https://mathoverflow.net/users/437 | 223546 | 104,658 |
https://mathoverflow.net/questions/223551 | 4 | A character of $U(\infty)$ is a continuous, positive definite class function $\chi: U(\infty) \longrightarrow \mathbb{C}$, which is normalized by $\chi(e) = 1$. Observe that the set of characters of $U(\infty)$ is a convex set, so we can talk about its extreme points, called the extreme characters of $U(\infty)$.
It ... | https://mathoverflow.net/users/nan | Conceptual explanation for multiplicativity of theta generalization of extreme characters of U(infty) | The extreme characters of $U(\infty)$ correspond to pure traces on a C\*-algebra associated to the group. Its (pre-ordered) $K\_0$ group admits a ring structure (in this case, via $U(\infty) \times U(\infty) \to U(\infty)$, which induces a tensor product-like ring structure on the $K\_0$-group). The pure traces are exa... | 1 | https://mathoverflow.net/users/42278 | 223559 | 104,663 |
https://mathoverflow.net/questions/223560 | 7 | For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG,
$$
$f(\xi)\in [B;BG]$, and the composition up to homotopy
$$
g(\xi): B\overset{f}{\longrightarrow} BG\overset{i}{\longrightarrow}BO(n)\overset{j... | https://mathoverflow.net/users/76736 | classifying maps of Whitney sums of vector bundles | $BO(n)$ is the infinite-dimensional Grassmannian $Gr(n,\infty)$ of $n$-planes in ${\mathbf R}^\infty$. There is a natural direct sum operation
$$\oplus\colon Gr(n,\infty)\times Gr(m,\infty)\to Gr(n+m,\infty)$$
(just taking the direct sum of linear subspaces) and it gives you the desired map
$$BO(n)\times BO(m)\to BO(n... | 9 | https://mathoverflow.net/users/39082 | 223562 | 104,664 |
https://mathoverflow.net/questions/223557 | 0 | In algebraic geometry, is the dual of the dual curve the original curve? Why or why not?
| https://mathoverflow.net/users/82765 | Is the dual of the dual curve the original curve? | This is claimed and proved as Theorem 1 here <https://math.berkeley.edu/~monks/papers/DualityV3.pdf>
| 1 | https://mathoverflow.net/users/5516 | 223563 | 104,665 |
https://mathoverflow.net/questions/223561 | 2 | Let $X$ be a smooth projective surface over $\mathbb{C}$. We know that nef line bundles form a closed convex cone whose interior is the ample cone. My doubt is the other direction, is it possible to have an ample class $C$ such that $C=C\_1+C\_2$ where neither $C\_i$ is nef? In fact I know that $C\in |L|$ where $L$ is ... | https://mathoverflow.net/users/70211 | Can an ample class be written as a sum of two classes which are not even nef? | Take an arbitrary surface which has lots of non-nef effective curves. In particular, take one that has two non-nef effective curves on "opposite" sides of the cone of effective curves. Then some linear combination of these two curves will lie in the interior of the cone and will be ample.
In particular, take a K3 sur... | 4 | https://mathoverflow.net/users/10076 | 223567 | 104,667 |
https://mathoverflow.net/questions/223574 | 5 | Let $k$ be a fixed positive integer and $\Sigma\_k$ the $k$-th symmetric group. By letting $\Sigma\_k$ permuting an orthonormal basis of a $k$-dimensional Euclidean space, there is a "regular representation"
$$
r\_k: \Sigma\_k\longrightarrow O(k)
$$
which induces a map between classifying spaces
$$
\rho\_k: B\Sigma\_k... | https://mathoverflow.net/users/76736 | how to prove the $n$-times self-product of a map is null-homotopic | Yes, this is true.
Your map $B\Sigma\_k\rightarrow BO(k)$ gives rise to a map $B\Sigma\_k\rightarrow BO$, i.e. an element in the reduced K-theory group $\tilde{ko}^0(B\Sigma\_k)$.
Now that whole group is probably not torsion, but the Atiyah-Hirzebruch spectral sequence tells you that, if $K$ is a finite-dimensional $... | 5 | https://mathoverflow.net/users/39747 | 223578 | 104,671 |
https://mathoverflow.net/questions/222705 | 4 | Let
$$X\_t = m \cdot t + W\_t$$
where $W\_t$ is a Brownian motion. Let
$$Z = \sup \{ t\in [0,1] : X\_t = 0\}.$$
It is known that if $m = 0$ then the distribution of $z$ is given by
$$\mathbb{P}[Z \leq y ] = \frac{2}{\pi} \arcsin \left( \sqrt{ y } \right).$$
What is the distribution of $z$ for general values of $m$?
... | https://mathoverflow.net/users/23661 | Arcsine law for Brownian motion with drift | Here is a simple description:
$$Z\stackrel{d}{=}A\cdot Y\_m$$ i.e. $Z$ is distributed as the product of two random variables, where the the factors are independent, $A$ is $\arcsin$-distributed, and $Y\_m$ is distributed as $\min\{1,\exp(m^2/2)\}$.
($\exp(\lambda)$ denotes an exponentially distributed rv. with exp... | 1 | https://mathoverflow.net/users/48831 | 223594 | 104,677 |
https://mathoverflow.net/questions/223588 | 9 | Given a nontrivial left ideal $J$ of a unital $C^\*$ algebra $A$, is there a state on $A$ which vanishes on all elements of $J$? (Left or right doesn't matter, just not 2-sided.)
The problem came from the idea of a state as evaluation at a 'point' of a noncommutative space. If an ideal corresponds to a vanishing 'se... | https://mathoverflow.net/users/29625 | States and left ideals | Yes if you restrict to positive elements. This is because a closed subset of a unital C\*-algebra is a left ideal if and only if it is a left kernel of a certain state. This was first observed by Prosser:
>
> R. T. Prosser, On the ideal structure of operator algebras, *Mem. Amer. Math. Soc.* **45** (1963).
>
>
> ... | 8 | https://mathoverflow.net/users/15129 | 223597 | 104,678 |
https://mathoverflow.net/questions/223595 | 2 | Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show
$$(X\mid X \geq 1) \succeq\_{sd} 1 + Y.$$
Here $(X \mid \cdot)$ is the conditioned random variable and $\succeq\_{sd}$ denotes stochastic dominance, formally:
$$\mathbf P[X \geq k \mid X \geq ... | https://mathoverflow.net/users/52896 | Conditioned binomial dominates unconditioned with different parameter | This does not seem to be true. Take $p=2/3$ and $n=k=1000$. Then, you need that
$$
\frac{(2/3)^{1000}}{1-(1/3)^{1000}} \geq (2/3)^{999},
$$
which is clearly false.
| 2 | https://mathoverflow.net/users/81488 | 223598 | 104,679 |
https://mathoverflow.net/questions/211150 | 2 | Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}\_{g,n}$ has dimension at least two the locus in $\overline{M}\_{g,n}$ parametrizing reducible curves which are union of an elliptic tails with $6$ automorphisms and a curve of genus $g-1$ is a locus of non canonical singularities? Could it happen that fo... | https://mathoverflow.net/users/nan | Non canonical singularities of moduli spaces of curves | We denote by $\Delta\_{1}$ the image in $\overline{M}\_{g,n}$ of $\overline{M}\_{1,1}\times \overline{M}\_{g-1,n+1}$, the divisor parametrizing curves with elliptic tails.
Let $Z\_6$ the image of $[E\_6]\times \overline{M}\_{g-1,n+1}\subset \Delta\_{1}$, the codimension two loci where the elliptic tail has six autom... | 0 | https://mathoverflow.net/users/14514 | 223606 | 104,680 |
https://mathoverflow.net/questions/223603 | 2 | I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on hyperbolic surfaces, but I was wondering if this statement can be made rigorous. In particular:
1) Can you point me out the... | https://mathoverflow.net/users/82786 | Lamination as limit of arcs |
>
> 1) Can you point me out the rigorous definition of "limit of a sequence of arcs" on a surface?
>
>
> 2) In which topological space are we considering the limit of arcs to obtain a lamination?
>
>
>
Suppose that $S$ is a orientable surface of finite topological type, equipped with a hyperbolic metric of fin... | 3 | https://mathoverflow.net/users/1650 | 223607 | 104,681 |
https://mathoverflow.net/questions/222568 | 8 | Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral:
$$I\_f(t):=\int\_{\mathbb{R}^3}\int\_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$
I need to show that $I\_f(t)$ is finite, and to find how fast it grows as $t\rightarrow +\infty$.
If $f$ decays fast enough at infinity so that $f\in L^1(\ma... | https://mathoverflow.net/users/54552 | Convergence of an oscillatory integral | Here is a more detailed version of my sketch in the comments above.
First of all, let $\rho$ be a compactly supported smooth function (cutoff function) such that $\rho=1$ on $B\_1$. Write $f=\rho f+(1-\rho)f$. Since $\rho f$ is compactly supported, it is $L^1$, and your old argument applies to it, so it remains to bo... | 2 | https://mathoverflow.net/users/37103 | 223612 | 104,682 |
https://mathoverflow.net/questions/223610 | 1 | We call a finite, simple, undirected graph $G=(V,E)$ an $n$-*Erdos graph* if there are $n$ subsets $S\_1,\ldots, S\_n$ of $V$ such that
1. $V = \bigcup\_{n=1}^n S\_n$;
2. each $S\_k$ has $n$ elements for $k\in\{1,\ldots, n\}$;
3. $\{a,b\}\in E$ implies that there is $k\in\{1,\ldots,n\}$ such that $a, b\in S\_k$;
4. $... | https://mathoverflow.net/users/8628 | Weak Erdos graphs | A complete graph with $\Theta(n^{3/2})$ vertices can be a weak $n$-Erdos graph.
Let $m$ be the largest integer satisfying $m(m-1)/2 \le n$. Consider $m$ disjoint copies of sets $V\_1, \dots, V\_m$ each with $\lfloor n/2 \rfloor$ elements. Let the sets $S\_i$s be
$$\{S\_1, S\_2, \dots, S\_{m(m-1)/2}\} = \{V\_1 \cup V... | 2 | https://mathoverflow.net/users/36579 | 223615 | 104,683 |
https://mathoverflow.net/questions/223591 | 4 | I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)\_{ \mathbb{Q} } = \displaystyle\bigoplus\_{ i + j = k } \mathrm{CH}^{i,j} (X) $. I don't remember exactly where i found that article on the net. Can anyone help me please ?
Thanks a lot to all o... | https://mathoverflow.net/users/82782 | About the decomposition of a Chow group of a variety | abx has basically answered your question in a comment, but let me add some detail. There is no natural such direct sum decomposition; you are probably thinking of the Bloch-Beilinson *filtration* on Chow groups, whose existence in general is only conjectural. A very nice reference is a paper by Jannsen in one of the "M... | 4 | https://mathoverflow.net/users/1310 | 223619 | 104,684 |
https://mathoverflow.net/questions/223631 | 1 | In the paper [Hernandez and Leclerc - Cluster algebras and quantum affine algebras](http://arxiv.org/abs/0903.1452), Section 13.5, it is said that when $\mathfrak{g}$ is of type $A\_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}\_2$ for $U\_q(\widehat{\mathfrak{g}})$ is of finite cluster type $D\_... | https://mathoverflow.net/users/11877 | Number of cluster variables | For $A\_k$ of level $\ell$, the cluster types are given by square grids of size $k \times \ell$.
Therefore the types you ask for are $E\_6$ and $E\_8$ (see [Scott - Grassmannians and cluster algebras](https://doi.org/10.1112/S0024611505015571)), for which you can easily find the
number of clusters using the usual for... | 2 | https://mathoverflow.net/users/10881 | 223638 | 104,691 |
https://mathoverflow.net/questions/223646 | 3 | I know that, by D. McKay and C. E. Praeger papers" Vertex-transitive graphs which are not Cayley graphs I", there exist 112 non-Cayley vertex-transitive graph with 24 vertices.
Is there any such graph of valency 5? Which graph is the answer?
| https://mathoverflow.net/users/27831 | Is there any vertex-transitive non-Cayley graph with 24 vertices and valency 5? | Yes, there is one, see [this paper](https://cs.anu.edu.au/people/Brendan.McKay/papers/Transitive26.pdf).
You can fetch it from [Gordon Royle's collection](http://staffhome.ecm.uwa.edu.au/~00013890/remote/trans/index.html).
| 4 | https://mathoverflow.net/users/9025 | 223648 | 104,693 |
https://mathoverflow.net/questions/219632 | 2 | Let $G$ be a Linear Algebraic Group (over algebraically closed field). We know that the connected component $G^o$ is a normal subgroup of finite index in $G$. Let $g$ be an element of $G$ which is not in $G^o$.
I want to understand Jordan decomposition of $g$ as a product of semisimple and unipotent elements.
Modi... | https://mathoverflow.net/users/69977 | Jordan decomposition of elements in non-connected component of algebraic group | The question itself falls somewhat short of being "research-level", but maybe it's useful to expand a little on the comment by anon (which the proposed "answer" by Anupam doesn't improve on).
The Jordan-Chevalley decomposition in an arbitrary linear (= affine) algebraic group $G$ over an algebraically closed field $... | 2 | https://mathoverflow.net/users/4231 | 223651 | 104,695 |
https://mathoverflow.net/questions/223658 | 2 |
>
> Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely
> generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ?
>
>
>
I know that this is true for Fuchsian groups and Bianchi groups. If this is unknown in general, then:
Are there any other families of Kleinian... | https://mathoverflow.net/users/38889 | Engulfing Kleinian groups? | First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroups. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamen... | 10 | https://mathoverflow.net/users/21684 | 223660 | 104,703 |
https://mathoverflow.net/questions/223634 | 5 | The Feichtinger's algebra
$S\_0(\mathbb{R^d})=M^{1,1}(\mathbb{R^d}):=\{f\in L^2(\mathbb{R^d}):V\_g(f)\in L^1(\mathbb{R^{2d}})\}$, where
$V\_g(f)(x,\omega)$ is the short-time Fourier transform of $f$ with respect to $g \in S(\mathbb{R^d})$, which is the Schwartz space, defined by
$$
V\_g(f)(x,\omega)=\int\_\mathbb{R^d} ... | https://mathoverflow.net/users/82802 | Is the Feichtinger's algebra $(S_0(\mathbb{R^d}),||\cdot||_{S_0})$ reflexive? | No. Wilson bases (or more generally local Fourier bases) allow to identify $S\_0$ with the sequence space $\ell^1$ (i.e. the elements of $S\_0$ are just the functions in $L^2$ having an absolutely convergent Wilson series expansion). HENCE the space is a dual space (the pre-dual is so to say $c\_0$ at the sequence spac... | 7 | https://mathoverflow.net/users/82814 | 223666 | 104,704 |
https://mathoverflow.net/questions/223662 | 2 | Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.
* a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
* a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^\*x)=0$ implies $x=0$; moreover $\tau(q)\leq 1$ for any projection $q\in M$).
Consider a projection $p\in M$, then its *central ... | https://mathoverflow.net/users/47294 | Approximation of the central support | I don't think your claim is true. As an easy counter example, just take $p=e\_{1,1}\in M\_n(\mathbb{C})$. Its central support is $1$, but it is minimal, so $z(p)$ is never a sum of projections dominated by $p$.
Perhaps you meant that each $p\_i \preceq p$, in the sense that there is a partial isometry $u$ such that $... | 3 | https://mathoverflow.net/users/351 | 223671 | 104,706 |
https://mathoverflow.net/questions/223541 | 3 | In a [previous question on MO](https://mathoverflow.net/questions/176737/when-is-a-newton-basin-fractal-continuously-determined-by-the-roots-of-its-polyn) I mentioned that I had convinced myself of the following:
>
> When $f\_p(z) = z - p(z)/p'(z)$ and $p$ is a complex polynomial, the Julia set, $J(f\_p)$ is *not* ... | https://mathoverflow.net/users/16518 | Reference for instability of Newton basins of polynomials at "separation" of a multiple root | I think for multiple roots this is clear. At a multiple root $z\_0$, you have an attracting fixed point. Take a small disc $D$ around this root; then (as you note) for a suitable perturbation $\tilde{p}$ of your polynomial, there will be at least two different roots well inside the disc $D$. These are (super-)attractin... | 1 | https://mathoverflow.net/users/3651 | 223686 | 104,712 |
https://mathoverflow.net/questions/221046 | 15 | The classical synthetic notion of *projective plane* consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and any two distinct lines intersect in a unique point (plus some nondegeneracy assumptions). There are similar notions of... | https://mathoverflow.net/users/49 | Synthetic projective lines | Building on previous work by Paul Libois, and related to work by Libois'
student Jean van Buggenhaut from 1969, Francis Buekenhout considered and
solved this question in "Foundations of one Dimensional Projective Geometry based on Perspectivities" Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 43 (... | 8 | https://mathoverflow.net/users/82821 | 223688 | 104,713 |
https://mathoverflow.net/questions/223680 | 3 | Let $X$ be a smooth rational variety of dimension $n$. We have $\dim H^0(X,\Omega\_X^p) = \dim H^0(\mathbb{P}^n,\Omega\_{\mathbb{P}^n}^p)$ for any $p$. These are Hodge numbers. I know that we can not expect an equality for the sections of the sheaves $\Omega\_X^p(k)$.
However, is it true that if $p < n$ then $\dim H... | https://mathoverflow.net/users/nan | Birational Invariants | No, that is not true, even after the modifications. Begin with a Veronese surface $X = v\_2(\mathbb{P}^2) \subset \mathbb{P}^5$, i.e., $$ X = \{ [x\_0^2,x\_0x\_1,x\_0x\_2,x\_1^2,x\_1x\_2,x\_2^2] \in \mathbb{P}^5 | [x\_0,x\_1,x\_2]\in \mathbb{P}^2 \} = $$ $$\{ [y\_{2,0,0},y\_{1,1,0},y\_{1,0,1},y\_{0,2,0},y\_{0,1,1},y\_{... | 3 | https://mathoverflow.net/users/13265 | 223696 | 104,716 |
https://mathoverflow.net/questions/223697 | 2 | What is the minimal cardinality of a family of sets of real numbers, each of positive Lebesgue measure, such that every set of real numbers of positive Lebesgue measure contains some member of the family as a subset?
One may work in the Cantor space if desired, and chose compact sets as elements of the family. In par... | https://mathoverflow.net/users/2415 | What is the cofinality of the positive measure sets of reals? | It is equal to cofinality of the null ideal which is also equal to the density of the measure algebra. See theorem 3.2.2 in Bartozynski Judah book.
| 6 | https://mathoverflow.net/users/82830 | 223710 | 104,719 |
https://mathoverflow.net/questions/223718 | 6 | I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ x^2+y^2-az^2=0, $$
with $|x|,|y|,|z|<T$ is $C(a) T \log T$, where $C(a)$ is a constant depending only on $a$. I would very muc... | https://mathoverflow.net/users/3635 | The number of integral solutions to $x^2+y^2-az^2=0$ | You need some further assumptions on $a$ to guarantee the existence of a solution to this equation, as there might not be solutions modulo some prime, say. The theorem of Hasse and Minkowski tells you exactly when solutions exist.
But anyway, the result you want is Theorem 8 in the paper:
Heath-Brown - A New Form of ... | 8 | https://mathoverflow.net/users/5101 | 223723 | 104,722 |
https://mathoverflow.net/questions/223623 | 1 | Let $X$ be a separable Banach space and $1<p<\infty$. We say that a sequence $(x\_{n})\_{n}$ in $X$ is weakly $p$-convergent to $x\in X$ if the sequence $(x\_{n}-x)\_{n}$ is weakly $p$-summable. A subset $K$ of $X$ is said to be relatively weakly $p$-compact if $K$ is contained in $S(B\_{l\_{p^{\*}}})$ for some operato... | https://mathoverflow.net/users/41619 | A characterization of relatively weakly $p$-compact sets | The answer is no even for separable $X$. Let $X=(\sum\_{n=1}^\infty \ell\_1^n)\_{p^\*}$. $B\_X$ has your $p$-subsequence property (same argument as for the unit ball of $\ell\_{p^\*}$). It is not relatively weakly $p$-compact because $X$ is not isomorphic to a quotient space of $\ell\_{p^\*}$.
| 1 | https://mathoverflow.net/users/2554 | 223736 | 104,724 |
https://mathoverflow.net/questions/223715 | 4 |
>
> For which $3$-manifolds $M$ is the fundamental group $\pi\_1(M)$
> finitely generated and has positive rank gradient?
>
>
>
Recall that the rank gradient of a finitely generated group $G$ is defined to be $$\inf\_{H} \frac{d(H) - 1}{[G : H]}$$ where the infimum is taken over all finite index subgroups $H$ o... | https://mathoverflow.net/users/38889 | Which 3-manifolds have positive rank gradient? | This is answered in the proof of Theorem 8.5 of [this paper](https://math.berkeley.edu/sites/default/files/faculty/files/virtualspecialICM.pdf). This
says that the rank gradient is zero iff $M$ is prime or $\mathbb{RP}^3\#\mathbb{RP}^3$.
**Edit:** There's a small step missing from the argument. The argument shows th... | 7 | https://mathoverflow.net/users/1345 | 223737 | 104,725 |
https://mathoverflow.net/questions/223717 | 1 | Consider a function $f(x)=g(x,h(x))$, which we know has a unique root $x^\*$. The functions $f$, $g$ and $h$ are all continuous in $x$ and behave nicely. Iteratively solving $g(x\_{i+1}, h(x\_i))=0$ with some starting guess $x\_0$, we are able to find $x^\*$.
My goal is to prove that this iteration indeed works. Can ... | https://mathoverflow.net/users/44464 | Iterative Root Finding | Slightly generalizing, we may state conditions in terms of $G(x,y):=g(x,h(y))$. For instance:
>
> Let $G$ be a continuous function on $\mathbb{R}^2$. Assume that the
> equation $G(x,x)=0$ has a unique solution $x^\*$ and that for some $M$
> any solution of $G(x,y)=0$ verifies $y\le x\le M$. Then any sequence
> ... | 1 | https://mathoverflow.net/users/6101 | 223738 | 104,726 |
https://mathoverflow.net/questions/223746 | 3 | [Euler function](https://en.wikipedia.org/wiki/Euler_function) is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod\_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
$$\phi(x)<\prod\_{i=1}^2(1-x^i)=1-x-x^2+x^3$$
>
> **What lower bounds are known for $\phi(... | https://mathoverflow.net/users/47499 | Lower bound for Euler's function | [mpmath](http://docs.sympy.org/dev/modules/mpmath/functions/qfunctions.html) can compute it very efficiently.
The function is called `mpmath.qp`. Here is a sage session for $\phi(e^{-\pi})$
with precision 100 decimal digits which takes about 360 microseconds on and old machine:
```
sage: time mpmath.qp(mpmath.exp(... | 2 | https://mathoverflow.net/users/12481 | 223748 | 104,727 |
https://mathoverflow.net/questions/223622 | 2 | I'm interested in the representation theory of the non-compact real Lie group $\mathrm{SO}^\*(2n)$, the subgroup of matrices $M\in\mathrm{SO}(2n,\mathbb{C})$ satisfying
$$
M^\dagger\eta \,M=\eta,\qquad \eta =
\begin{pmatrix}
0&\mathbb{I}\_n \\
-\mathbb{I}\_n &0
\end{pmatrix}.
$$
Due to its peculiar name, it is extrem... | https://mathoverflow.net/users/75321 | reference on representation theory of SO*(2n) | The Atlas of Lie Groups and Representations software gives copious explicit information about the representation theory of real reductive groups, including $SO^\*(2n)$. The software is freely available from <http://www.liegroups.org>. (It is currently under development and not yet well documented.) There is a web-based... | 6 | https://mathoverflow.net/users/6030 | 223760 | 104,732 |
https://mathoverflow.net/questions/223756 | 4 | In Kolar, Michor, & Slovak's book *Natural Operations in Differential Geometry*, it is proved the exterior derivative is universal in the following sense.
**Proposition 25.4.** For $k>0$ all natural operators $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$ are constant multiples of the exterior derivative.
... | https://mathoverflow.net/users/69037 | Exterior derivative as only (up to multiple) natural operator $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$ | Naturality here means $f^\*d\phi = d f^\*\phi$ for any smooth map, diffeomorphism or not. This is formally stronger than requiring this just for diffeos. Eventually the two conditions are equivalent, after checking the results.
Edit:
=====
Answer to Arrows comment.
If you require commuting only with pullbacks wi... | 6 | https://mathoverflow.net/users/26935 | 223766 | 104,735 |
https://mathoverflow.net/questions/223424 | 15 | The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat} \to \mathsf{Cat}$). This is equivalent to $\mathbb{Z}/2$ considered as a locally discrete 2-groupoid: the terminal obj... | https://mathoverflow.net/users/2362 | Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences? | $\newcommand{\hoCat}{\mathrm{ho}\mathsf{Cat}}
\newcommand{\Cat}{\mathsf{Cat}}
\newcommand{\Gaunt}{\mathsf{Gaunt}}
\newcommand{\hoGaunt}{\mathrm{ho}\mathsf{Gaunt}}
\newcommand{\Free}{\mathsf{Free}}
\newcommand{\hoFree}{\mathrm{ho}\mathsf{Free}}
\newcommand{\comp}{\mathrm{comp}}
\newcommand{\inv}{^{-1}}
\require{AMScd}$
... | 7 | https://mathoverflow.net/users/2362 | 223773 | 104,739 |
https://mathoverflow.net/questions/223763 | 6 | Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$.
Is it known when $Aut(X)$ is finite ?
Does there exist a formula for the dimension of the tangent space to the space of first order infinitesimal deformations $H^1(X,T\_X)$ ?
I am especially interested in the case of a codimension... | https://mathoverflow.net/users/nan | Automorphisms and infinitesimal deformations of a smooth complete intersection | It is known that there exists complete intersection of given type $T=(d\_1,\ldots,d\_c;n)$ with infinite automorphism group if and only if the type satisfies one of the following:
$$T \in \{(2;n), (3;1), (2,2;1), (4;2), (2,3;2), (2,2,2;2) \}.$$
I.e. quadrics, curves of genus $1$ and K3 surfaces. In all other cases, the... | 8 | https://mathoverflow.net/users/5101 | 223777 | 104,741 |
https://mathoverflow.net/questions/223790 | 5 | Let $A(k)$ be a $k$-algebra with two generators, $x$, $y$, and one defining relation: $yx - xy = 1$. What is the center of the algebra $A(k)$ in the case $\text{char}\,k > 0$?
| https://mathoverflow.net/users/nan | Center of $k$-algebra with two generators and sole defining relation $yx - xy = 1$ when $\text{char}\,k > 0$ | This is the [Weyl algebra](https://en.wikipedia.org/wiki/Weyl_algebra). The center is $k[x^p,y^p]$, where $p$ is the characteristic. See, for example, [here](https://math.stackexchange.com/questions/129834/) for a proof.
| 14 | https://mathoverflow.net/users/297 | 223791 | 104,747 |
https://mathoverflow.net/questions/223270 | 19 | The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external properties of the model, the simplest of which is cardinality.
**Question:** Suppose there is an uncountable transiti... | https://mathoverflow.net/users/1058 | Removing large cardinals from an uncountable transitive model | I think the answer is (consistently) no.
Following Asaf's comment, let $\kappa$ be Mahlo in $L$ and let $G \subset \text{Col}(\omega,\mathord{<}\kappa)$ be an $L$-generic filter. Let $\eta$ be the least ordinal such that $\kappa < \eta$ and $L\_\eta \models \mathsf{ZFC}$, and consider the model $N = L\_\eta[G]$.
Th... | 18 | https://mathoverflow.net/users/1682 | 223794 | 104,748 |
https://mathoverflow.net/questions/223796 | 4 | Doubly degenerate surface groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows:
Take a closed surface $\Sigma$ of genus at least $2$ and a pseudo-Anosov map $\varphi:\Sigma\to\Sigma$ and form the ma... | https://mathoverflow.net/users/82869 | Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus | Take any pair of measured laminations $\lambda,\mu$ which each fill the surface and are transverse to each other. Take sequences $\sigma\_i,\tau\_i$ in Teichmuller space, such that $\sigma\_i$ converges to $\lambda$ and $\tau\_i$ converges to $\mu$ in compactified Teichmuller space. Thurston's double limit theorem says... | 8 | https://mathoverflow.net/users/20787 | 223797 | 104,749 |
https://mathoverflow.net/questions/223377 | 10 | As a follow-up of Allen's question [Coxeter exchanges in non-reduced words](https://mathoverflow.net/questions/223129), I wonder whether it is known that the Demazure product is well-defined in Artin groups. This is:
Let $(W,S)$ be a Coxeter system with corresponding Artin system $A(W)$ which is the group of words in... | https://mathoverflow.net/users/21291 | Demazure product in Coxeter and Artin groups | A complete combinatorial proof using Allen's comment:
Let $(W,S)$ be a Coxeter system, and let $Dem(T) \in W$ be the **Demazure product** or **greedy product** of a word $T$ in $S$.
**Claim:** $Dem(T)$ is the unique Bruhat maximal element in the set $\big\{ \prod Q : Q \subseteq T\big\}$ (where $Q\subseteq T$ means... | 2 | https://mathoverflow.net/users/21291 | 223818 | 104,759 |
https://mathoverflow.net/questions/209732 | 29 | Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called "$\Pi$" because tensoring with it is "parity reversal".)
Indeed, there is a good notion of "direct sum of categories" in which case... | https://mathoverflow.net/users/78 | Is there a symmetric monoidal 2-category "SuperDuperVect"? | Yes, the symmetric monoidal 2-category you are looking for does exist.
I think that there is a slightly different 2-category which is better, but yours embedds inside the one I will describe, which differs in that
there are interesting "cross-terms" from $\Xi$ to 1, i.e. morphisms between these objects. This 2-catego... | 16 | https://mathoverflow.net/users/184 | 223826 | 104,763 |
https://mathoverflow.net/questions/223825 | 6 | I'm struggling to make sense of the Hodge star as a global canonical object. Here are my struggles so far and some questions:
Let $M$ be a finitely generated projective $R$-module (hence locally free and finitely presented). Suppose the $M$ is of rank $n$ and is equipped with a nondegenerate symmetric bilinear form $... | https://mathoverflow.net/users/22810 | A systematic canonical construction of the Hodge star operator | The first step is to construct a pairing on the modules $\bigwedge^k M$. I will assume that the pairing $g:M\otimes M\to R$ is perfect, that is it induces an isomorphism $M\to \textrm{Hom}(M,R)$. Then we can define a pairing on $\bigwedge^k M$ as follows
$$g(v\_1\wedge\cdots\wedge v\_k, w\_1\wedge \cdots\wedge w\_k) ... | 7 | https://mathoverflow.net/users/43054 | 223828 | 104,764 |
https://mathoverflow.net/questions/223785 | 3 | Let $V$ be a complex vector spaces and assume that a compact group G acts linearly on $V$. Then look at the $G$-equivariant polynomial maps from $V$ to $V$. Denote this by $Mor\_G(V,V)$. In the case of finite groups, $Mor\_G(V,V)$ is a finitely generated module over the ring of polynomial invariants $\mathbb{C}[V]$. Do... | https://mathoverflow.net/users/76018 | Equivariant polynomial maps | The answer should be yes, by the following argument:
With $n := \dim(V)$, consider the polynomial ring $C[x\_1..x\_n,y\_1..y\_n] = C[V \oplus V^\*]$, so G acts on the $x\_i$ as on the variables of $C[V]$, but on the $y\_i$ as on the variables of $C[V^\*]$ with $V^\*$ the dual space.
$C[x\_1..x\_n,y\_1..y\_n]^G$ is ... | 9 | https://mathoverflow.net/users/82616 | 223838 | 104,768 |
https://mathoverflow.net/questions/223810 | 21 | Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number is rational?
Edit: The system of equations and inequalities may have $n$ real-valued variables, and the real number to ... | https://mathoverflow.net/users/56920 | Is being rational decidable? | This is a long comment rather than an answer.
Your problem is what is known as a "promise problem." The input comes equipped with a promise, namely that there exists a unique real solution. Presumably, if the promise is violated, then the algorithm is "off the hook" and can do anything it likes, including failing to ... | 8 | https://mathoverflow.net/users/3106 | 223845 | 104,769 |
https://mathoverflow.net/questions/223847 | 3 | Fix $d \in \mathbb{N}$. Do there exist mutually disjoint connected open sets $V\_1,\ldots,V\_n \subset \mathbb{R}^d$ and $\mathbf{v} \in \mathbb{R}^d$ such that
* $\mathbb{R}^d \setminus (\bigcup\_{i=1}^n V\_i)$ is a Lebesgue-null set, and
* for each $i \in \{1,\ldots,n\}$, for every $\mathbf{x} \in V\_i$, $\,\mathbf... | https://mathoverflow.net/users/15570 | Existence or otherwise of a set of "sufficiently intricate" open sets | The answer is "yes" for $d\ge 3$ and "no" for $d\le 2$.
For $d=3$, let $V\_1$, $V\_2\subset\mathbb R^3$ be the open interiors of the sets
$$U\_1=\{\,(x\_1,x\_2,x\_3)\mid\lfloor x\_i\rfloor\text{ is even for at most one }i\,\}\;,$$
$$U\_2=\{\,(x\_1,x\_2,x\_3)\mid\lfloor x\_i\rfloor\text{ is even for at least two }i\,\... | 5 | https://mathoverflow.net/users/70808 | 223850 | 104,770 |
https://mathoverflow.net/questions/223716 | 1 | Let $W$ and $S$ are two positive valued continuous random variable. Suppose
$g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ is a constant.
Is it possible, there exist some $\theta>0$ such that the following two statements holds simultaneously,... | https://mathoverflow.net/users/62190 | Independence of two random variable | Take any real $\theta>0$ and any independent positive continuous random variables $W$ and $X$. Let $S:=X/(W+\theta)$ and $g(x):=\theta$ for all $x$. Then $SW+S\theta=SW+Sg(S)=X$, so that $W$ is independent of $SW+S\theta$ and of $SW+Sg(S)$.
| 2 | https://mathoverflow.net/users/36721 | 223851 | 104,771 |
https://mathoverflow.net/questions/223782 | 3 | Choose a notion of an "ambient homotopy" between maps of topological spaces. For example, say that two embeddings $Y \rightarrow X$ are ambiently homotopic if there is a path between them in the space of embeddings from $Y$ to $X$.
Q. Given a fiber bundle $p: E \rightarrow B$, and two ambiently homotopic maps $f, g :... | https://mathoverflow.net/users/39004 | "Ambient homotopy" between preimages under a fiber bundle? | The comment above was a bit rushed, sorry. Let $H \colon X \times [0,1] \to B$ be a homotopy between $f$ and $g$ and consider the pullback $(X \times [0,1])\_{H}$ of the bundle $p$ along $H$. The pullback comes with a map
$$ (X\times[0,1])\_{H} \xrightarrow{\phi} E $$
covering the homotopy $H$. If for each $t$, the ma... | 4 | https://mathoverflow.net/users/33141 | 223863 | 104,778 |
https://mathoverflow.net/questions/223843 | 1 |
>
> **Definition:** Very well covered graph to be a well-covered graph (possibly disconnected, but with no isolated vertices) in which each
> maximal independent set (and therefore also each minimal vertex cover)
> contains exactly half of the vertices.
>
>
>
Let $G$ be a very well covered graph.
**Prove\dis... | https://mathoverflow.net/users/68302 | Properties of very well covered graph | Is the following graph not a counter example?
Consider a graph on 6 vertices $u$, $v$, $w$, $u'$, $v'$, $w'$ with the following structure: $i)$ $u, v, w$ form a triangle, and $ii)$ $u'$ is connected to $u$, $v'$ to $v$ and $w'$ to $w$.
Each maximal independent set appears to contain $3=|V|/2$ vertices, and $G$ con... | 3 | https://mathoverflow.net/users/62684 | 223866 | 104,779 |
https://mathoverflow.net/questions/223854 | 8 | Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction
$$\phi|\_{T}:T\to T$$
is the power map $t\mapsto t^{\ell}$.
Let $\mathscr{L}$ be a line bundle on $X$. Then its pushforwa... | https://mathoverflow.net/users/4721 | Pushforward of line bundle under "toric isogeny" | It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such a push-forward is a direct sum of line bundles. To be precise, his theorem applies to the Frobenius map on a smooth toric variety in characteristic $p$, but the same... | 11 | https://mathoverflow.net/users/3847 | 223868 | 104,781 |
https://mathoverflow.net/questions/127021 | 2 | We have that S5 modal logic is characterized by the modal axioms $K$, $M$ (reflexive), $4$ (transitive), and $B$ (symmetric). That is, an equivalence relation on a set of possible world (which can be thought of as a digraph if you don't like Kripke semantics) generates S5 modal logic. There is a result of Hardy and Ram... | https://mathoverflow.net/users/20343 | On the Combinatorial Classification of Modal Kripke Frames | I happened to end up writing a paper which answered my question if anyone else is ever interested in this topic or comes across this question:
"On Generalizing a Temporal Formalism for Game Theory to the Asymptotic Combinatorics of S5 Modal Frames" ([arXiv link](http://arxiv.org/abs/1305.0064))
| 0 | https://mathoverflow.net/users/20343 | 223871 | 104,782 |
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