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/13793 | 3 | Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are [chorded approximations](http://www.tinaja.com/glib/nubzlen1.pdf) and the opinion that the analytic solution is "unbearably gruesome". However, I believe if I had the solution to this elliptic integral: $\int\_0^u... | https://mathoverflow.net/users/3716 | Finding the length of a cubic B-spline | You should first accept the fact that it's an elliptic integral, and therefore doesn't have an elementary expression without elliptic functions. If you had a numerical library with elliptic functions, then great. Otherwise, you need to either implement elliptic functions yourself, or implement numerical integration of ... | 13 | https://mathoverflow.net/users/1450 | 13796 | 9,307 |
https://mathoverflow.net/questions/11675 | 9 | Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a *Borsuk pair* if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is **odd** on {$x$
$\in$ $X$ ; $||x||=1$}, it follows that the set $T^{-1}$( { 0 } )
is nonempty.
My conjecture is : $(X,\ Y)$ Borsuk pair $\... | https://mathoverflow.net/users/2508 | Borsuk pairs of Banach spaces | Without loss of generality you can think that $Y=\ell^1$ (just pick an infinite sequence of linearly independent unit vectors in $Y$ and multiply them by the coefficients decaying so fast that no non-trivial $\ell^1$ combination will have any chance to be $0$). Now, construct a locally finite cover of $X$ with opposite... | 14 | https://mathoverflow.net/users/1131 | 13798 | 9,308 |
https://mathoverflow.net/questions/13825 | 1 | I was reading [this paper](http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf) a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.
I'll reproduce the diagram:
$$\ \ \matrix{
&&0... | https://mathoverflow.net/users/1353 | The sharp 3x3 lemma: a proof by universal properties? | This should follow from the Salamander lemma, which, as you found out in your [previous MO question,](https://mathoverflow.net/questions/6749/a-proof-of-the-salamander-lemma-without-mitchells-embedding-theorem) can be proved without the use of elements. It is a nice exercise, so I won't spell it out for you.
| 3 | https://mathoverflow.net/users/184 | 13830 | 9,326 |
https://mathoverflow.net/questions/13839 | 14 | On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H\_k(X,\mathbb{Z})$ (assuming I've got all the indices right) called the $k$th intermediate Jacobian.
If $k=1$, we have $J^1(X)=H^{1,0}/H\_1$, and so $J^1(X)\cong Jac(X)$ is an abelian... | https://mathoverflow.net/users/622 | What do intermediate Jacobians do? | Probably you know this, but just to be sure: they receive cycle class maps from
codimension $k$ cycles. More precisely, if $Z$ is a cycle on $X$ of codimension $k$
that is cohomologically trivial, then it gives an element in $J^k(X)$. This map,
from cohomologically trivial cycles to $J^k(X)$, is a (generalized) Abel-Ja... | 14 | https://mathoverflow.net/users/2874 | 13841 | 9,333 |
https://mathoverflow.net/questions/13714 | 24 | Assume $\Gamma$ be a Bieberbach group which acts on $\mathbb R^n$
(i.e. a discrete subgroup of isometries of $n$-dimensional Euclidean
space with a compact fundamental domain).
Denote by $M(\Gamma)$ the number of maximal finite subgroups (up to
conjugation) in $\Gamma$.
Is it true that $M(\Gamma)\le 2^n$?
Things I ca... | https://mathoverflow.net/users/3700 | Number of subgroups in a Bieberbach group. | Dima, I can not write a comment (yet) so I will start an answer to my
own question.
You may assume that $\Gamma$ acts by isometries, so $A=\mathbb
R^n/\Gamma$ is an Alexandrov space.
For each maximal subgroup $F$ one can take its fixed point set $S\_F$
in $\mathbb R^n$.
$S\_F$ is an (affine) subspace and image (say $... | 11 | https://mathoverflow.net/users/3700 | 13846 | 9,336 |
https://mathoverflow.net/questions/13843 | 17 | We have a natural number $n>1$. We want to determine whether there exist
natural numbers $a, k>1$ such that $n = a^k$.
Please suggest a polynomial-time algorithm.
| https://mathoverflow.net/users/1735 | How to quickly determine whether a given natural number is a power of another natural number? | This can be done in "essentially linear time." Check out Daniel Bernstein's website: <http://cr.yp.to/arith.html>
Especially note his papers labeled [powers] and [powers2].
| 25 | https://mathoverflow.net/users/3199 | 13852 | 9,339 |
https://mathoverflow.net/questions/13844 | 5 | Suppose we have an arbitrary function $f : \mathbb{R}^2 \to \mathbb{R}$. For any subset $s \subseteq \mathbb{R}^2$, we can define $g\_f(s)$ as the integral\* of $f$ over the region $s$. Suppose further that we have access to an oracle that will tell us the value of $g\_f(s)$ for any $s$.
Now, restrict our attention t... | https://mathoverflow.net/users/2785 | Heaviest Convex Polygon | It should be polynomial (probably O(N^3)) in the number of input points using the dynamic programming technique in my paper with Overmars et al, "Finding minimum area k-gons", Disc. Comput. Geom. 7:45-58, 1992, [doi:10.1007/BF02187823](http://dx.doi.org/10.1007/BF02187823).
The idea is: for each three points p,q,r, l... | 5 | https://mathoverflow.net/users/440 | 13855 | 9,341 |
https://mathoverflow.net/questions/13850 | 1 | I was reading this question [link text](https://mathoverflow.net/questions/5772/principal-bundles-representations-and-vector-bundles)
and can't seem to see why, if $\pi: P \to B$ is a principle $G$-bundle and $$\rho:G \to GL\_n(\mathbb{C})$$ is a representation of $G$, then the total space $P \times\_{\rho} \mathbb{C}^... | https://mathoverflow.net/users/2612 | Local Triviality of an Associated Bundle | A principal G-bundle $\pi: P \to B$ is locally equivalent to a product. Depending on who you ask, this is either part of the definition, or a short lemma. It means that there is a cover of B by open sets U, together with bundle isomorphisms $\alpha\_U: \pi^{-1}U \to G \times U$ that are both G-equivariant, and induce t... | 3 | https://mathoverflow.net/users/121 | 13858 | 9,342 |
https://mathoverflow.net/questions/13851 | 71 | I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M\_{23}$, have been proven to be Galois groups over $\mathbb{Q}$.
In particular, the Monster group has been proven to be a Galo... | https://mathoverflow.net/users/1079 | The inverse Galois problem and the Monster | the monster is a nice example of how the so-called **rigidity method** for the inverse Galois problem works. There is a lot of beautiful mathematics behind this, I will sketch the different steps.
A general remark: it is known that every profinite group, i.e. every group which could be a Galois group of some field ex... | 86 | https://mathoverflow.net/users/1107 | 13861 | 9,343 |
https://mathoverflow.net/questions/13577 | 3 | Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims the following:
* $\xi f$ exists iff $\mu(|f| \wedge 1) < \infty$.
It's certainly a basic result on Poisson Random Mea... | https://mathoverflow.net/users/3676 | Non-existence of integral with respect to Poisson Random Measure | As I mentioned in my comment, you can prove the statement and its converse by looking at the moment generating function. Supposing that f ≥ 0 and λ > 0 is a real number, the following is true for a Poisson point measure ξ with Eξ = μ,
$$\mathbb{E}\left[e^{-\lambda\xi f}\right]=\exp\left(-\mu\left(1-e^{-\lambda f}\rig... | 5 | https://mathoverflow.net/users/1004 | 13878 | 9,349 |
https://mathoverflow.net/questions/13477 | 4 | An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is normal if and only if for all $g \in G(S)$ and $n \in \mathbb N \setminus \{0\}$, $ng \in S$ implies $g \in S$.
Let $... | https://mathoverflow.net/users/3649 | Normality of an affine semigroup | I worked on this for a bit with Ricky Liu, who came up with this very quick solution:
Take your set T. Suppose $\sum p\_i = kn$, where $k \geq 2$. Create the following set T': let $i$ appear $p\_i$ times. This creates a set $T'$ with at least $2n$ elements (by your first constraint and $k \geq 2$), whose sum is divis... | 4 | https://mathoverflow.net/users/81883 | 13879 | 9,350 |
https://mathoverflow.net/questions/13882 | 20 | Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on affine everything and will work out the general case at some future time.
The question that this got me thinking about... | https://mathoverflow.net/users/14672 | Closed vs Rational Points on Schemes | If $k$ is algebraically closed and $X$ is a $k$-scheme locally of finite type, then the $k$-rational points are precisely the closed points. (See EGA 1971, Ch. I, Corollaire 6.5.3).
More generally: if $k$ is a field and $X$ is a $k$-scheme locally of finite type, then $X$ is a Jacobson scheme (i.e. it is quasi-isomor... | 19 | https://mathoverflow.net/users/717 | 13886 | 9,355 |
https://mathoverflow.net/questions/13897 | 31 | In every group theory textbook I've read, the holomorph has been defined, and maybe a few problems done with it. I've also seen papers focusing on computing Hol($G$) for a specific class of $G$.
One thing I have never seen is any actual use for it. Are there major results using the holomorph of a group? Does it occur... | https://mathoverflow.net/users/2616 | Uses of the holomorph, Hol($G$) = $G \rtimes $ Aut($G$) | If G is abelian, then the holomorph of G is a reasonably nice group. If G is a finite elementary abelian p-group of order pn, then you can consider it to be a vector space over Z/pZ. The automorphism group is the group GL(n,p) of invertible n×n matrices over Z/pZ. The holomorph is called the affine general linear group... | 9 | https://mathoverflow.net/users/3710 | 13914 | 9,372 |
https://mathoverflow.net/questions/2757 | 24 | For positive integers $n$ and $L$, denote by $SL\_n(Z,L)$ the level $L$ congruence subgroup of $SL\_n(Z)$, i.e. the kernel of the homomorphism $SL\_n(Z)\rightarrow SL\_n(Z/LZ)$.
For $n$ at least $3$, it is known that $SL\_n(Z,L)$ is normally generated (as a subgroup of $SL\_n(Z)$) by Lth powers of elementary matrices... | https://mathoverflow.net/users/317 | Generators for congruence subgroups of SL_2 | Hi Andy,
I don't know if you are still interested in this, but I just found the reference:
MR0049937 (14,250d)
Grosswald, Emil
On the parabolic generators of the principal congruence subgroups of the modular group.
Amer. J. Math. 74, (1952). 435--443.
It is based on the previous work of H.Frasch (1933) who gave a... | 13 | https://mathoverflow.net/users/3738 | 13915 | 9,373 |
https://mathoverflow.net/questions/13909 | 25 | I'm a bit confused by the double role which sheaves play in the theory of stacks.
On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other hand a sheaf on a site is (or better its associated category fibered in sets is) a very particular stack itself, so a gene... | https://mathoverflow.net/users/828 | Stacks and sheaves | Let me see if I understand your example correctly: you are fixing $X$ and $Y$, families
of curves over $S$, and now you are considering the functor which maps an $S$-scheme $T$
to the set of $T$-isomorphisms $f^\*X \to f^\*Y$ (where $f$ is the map from $T$ to $S$).
If I have things straight, then this functor shouldn... | 21 | https://mathoverflow.net/users/2874 | 13917 | 9,375 |
https://mathoverflow.net/questions/10581 | 9 | There exist a large family of noncommutative spaces that arise from the quantum matrices. These algebraic objects $q$-deform the coordinate rings of certain varieties. For example, take quantum $SU(2)$, this is the algebra $< a,b,c,d >$ quotiented by the ideal generated by
$$
ab−qba, ~~ ac−qca, ~~ bc−cb, ~~ bd−qdb, ~~ ... | https://mathoverflow.net/users/1095 | Kontsevich, and Geometric, Quantization and the Podles sphere | As far as I understand, the flag manifolds with Kahler structures mentioned in the question are simply coadjoint orbits of compact Lie groups with the Kirillov-Kostant-Souriau bracket, so their quantizations will yield quotients of the usual enveloping algebra $U(g)$ and will not have to do with quantum groups. I suppo... | 6 | https://mathoverflow.net/users/3696 | 13920 | 9,377 |
https://mathoverflow.net/questions/9011 | 8 | Defining the translated Gaussians by $f\_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f\_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon>0$. As a consequence, low-frequency trigonometric functions are dense in $L^2([a,b])$. The proof of the first result uses ... | https://mathoverflow.net/users/674 | Approximating with translated Gaussians and low-frequency trig functions | Wiener's approximation theorem says that
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a\_i h(\cdot - x\_i): a\_i, x\_i \in \mathbb{R}\}$ is dense in $L^2(\mathbb{R})$ if and only if zeros of the Fourier transform of $h$ has zero Lebesgue measure.
See Wiener's book "The Fourier Integral and Certa... | 2 | https://mathoverflow.net/users/3736 | 13931 | 9,382 |
https://mathoverflow.net/questions/13934 | 5 | Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most I knew was that 3 is impossible and 7 is possible (tile hexagons of diameter 1-ε). I haven't heard about this problem s... | https://mathoverflow.net/users/27 | Coloring Points in the Plane | This is the [Hadwiger–Nelson problem](http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem).
| 9 | https://mathoverflow.net/users/1409 | 13936 | 9,385 |
https://mathoverflow.net/questions/13868 | 5 | This question arose after I thought about Ben Webster's comments to [this question](https://mathoverflow.net/questions/10532/).
There he asked me what was my definition of a moduli problem. When I came to think of it, I never saw a precise definition like that. My understanding is along the following lines.
Roughl... | https://mathoverflow.net/users/2938 | Proper definition of a moduli problem | Since not many people have had anything to say, I thought I might make a few remarks. But beware that this is all what I've passively picked up over the years---it's not the result of an actual study of things.
I think the right definition of a moduli problem is a fibered category $p:E\to B$. This should be thought o... | 7 | https://mathoverflow.net/users/1114 | 13960 | 9,393 |
https://mathoverflow.net/questions/13251 | 11 | **Background**
Yet another homework inspired question: A scheme is reduced if no section of the structure sheaf is nilpotent. To every scheme $X$ there is a scheme $X\_{red}$ and a morphism $i: X\_{red} \rightarrow X$ such that every morphism from a reduced scheme into $X$ factors through $X\_{red}$. Hartshorne Ex. 2... | https://mathoverflow.net/users/1106 | Endofunctors of CRing which give schemes when composed with schemes? | Here is a nontrivial example I like. Let $W:\mathrm{Rings}\to\mathrm{Rings}$ denote the Witt vector functor of some fixed finite length. (You can consider the $p$-typical Witt vectors, for some prime $p$, but everything works with the other standard flavors.) Then the functor $W\_\*(-)=-\circ W$ is an endofunctor of th... | 8 | https://mathoverflow.net/users/1114 | 13962 | 9,395 |
https://mathoverflow.net/questions/13942 | 11 | Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. The monoid structure I'm referring to is the connect sum of knots.
Bayer-Fluckiger has a result in particular that sa... | https://mathoverflow.net/users/1465 | Computing the structure of the group completion of an abelian monoid, how hard can it be? |
>
> Do people have good examples where it's "easy" to compute the group-completion of a commutative monoid, but for which the monoid itself is still rather mysterious?
>
>
>
This happens all the time in K-theory $K^0(X)$, both algebraic and topological. Perhaps it is even the reason that K-theory is a useful too... | 28 | https://mathoverflow.net/users/126667 | 13979 | 9,401 |
https://mathoverflow.net/questions/13989 | 9 | Suppose $E\_1$ and $E\_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.
But $E\_1$ and $E\_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E\_d$. Now the qu... | https://mathoverflow.net/users/3757 | Two questions on isomorphic elliptic curves | Question 1: Putting both curves in say, Legendre Normal Form (or else appealing the lefschetz principle) shows that if the two curves are isomorphic over $\mathbf{C}$ then they are isomorphic over $\overline{\mathbf{Q}}$. Now we could say that for instance $E\_2$ is an element of $H^1(G\_{\overline{Q}}, Isom(E\_1))$ wh... | 7 | https://mathoverflow.net/users/3384 | 13997 | 9,413 |
https://mathoverflow.net/questions/13995 | 12 | First of all, I know the concepts of isomorphism and equivalence between categories, and that the latter one is the more interesting one, whereas the first is rather rare and uninteresting.
Are there isomorphisms of categories, which are not trivial and not pathological? I regard the examples on [wikipedia](http://e... | https://mathoverflow.net/users/2841 | nontrivial isomorphisms of categories | Whether this counts as trivial is a subjective matter, but here goes.
Any adjunction
$$
F: C \to D,\ \ \ G: D \to C
$$
(with $F$ left adjoint to $G$) gives rise canonically to a monad $T = GF$ on $C$ and a "comparison" functor $K: D \to C^T$. Here $C^T$ is the category of algebras for the monad $T$. The adjunction ... | 16 | https://mathoverflow.net/users/586 | 14004 | 9,417 |
https://mathoverflow.net/questions/13990 | 19 | Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles.
So my questions are the following:
1) what is an evidence for this conjecture?
2)why is the condition on $n\geq 7$, but not oth... | https://mathoverflow.net/users/2348 | Evidences on Hartshorne's conjecture? References? | This [answer](https://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689) of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to a... | 4 | https://mathoverflow.net/users/2083 | 14005 | 9,418 |
https://mathoverflow.net/questions/14003 | 4 | Consider a scaled sine function, $\sin(2\pi x/2^n)$, for some positive integer $n$. For this, I have the following linear combination.
$$ \sum\_{x=1}^{2^{n-2}} c\_x \sin(2\pi x/2^n).$$
(The upper limit to the sum is $2^{n-2}$.)
The question is whether there exist $c\_x \in \{0, \pm 1, \pm 2\}$, not all $0$, that ma... | https://mathoverflow.net/users/3760 | A sum involving sines | No, such $c\_x$ don't exist. Even if you replace $\lbrace 0,\pm 1,\pm 2\rbrace$ by $\mathbb Q$, this won't change. In fact, if they would exist, then, using the relation $\displaystyle \sin\frac{2\pi x}{2^n} = \frac{\zeta^x-\zeta^{-x}}{2i}$ (where $\zeta$ is a primitive $2^n$-th root of unity), the equation
$\displayst... | 15 | https://mathoverflow.net/users/2530 | 14006 | 9,419 |
https://mathoverflow.net/questions/14001 | 2 | Let $z\_1,\ldots,z\_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k\_j\in\mathbb{N}$ such that $\lim\_{j\to\infty}z\_i^{k\_j}=1$ for all i?
| https://mathoverflow.net/users/1626 | Simultaneous convergence of powers of unit complex numbers | Yes. Let $z=(z\_1,...,z\_n)$, a point on the torus $(S^1)^n$. Since the torus is compact, the sequence $z^m$ has a convergent subsequence $z^{m\_1}$, $z^{m\_2}$,..., and we may choose it so that $k\_j=m\_{j+1}-m\_j$ are nonnegative and increasing. Then ${\rm lim} z\_i^{k\_j}=1$ for all $i$.
| 7 | https://mathoverflow.net/users/3696 | 14008 | 9,420 |
https://mathoverflow.net/questions/14010 | 3 | I've been trying to understand principal bundles, and to that end have been looking at the bundle
$$
\pi: SU(2) \to \mathbb{CP}^1,~~~ (a\_{ij}) \mapsto [a\_{11},a\_{21}],
$$
with fibre $U(1)$. I assumed that the bundle would be trivial over the standard nbds $U\_1,U\_2 \subset \mathbb{C}$, but can't seem to identify t... | https://mathoverflow.net/users/2612 | Transition Functions of the Principal Bundle $SU(2) \to \mathbb{CP}^1$ | You have two charts $U\_0=\Bbb C$ and $U\_\infty=(\Bbb C\setminus 0)\cup \infty$.
The transition function on the intersection $\Bbb C\setminus 0$ is $g(z)=z/|z|$.
| 4 | https://mathoverflow.net/users/3696 | 14013 | 9,423 |
https://mathoverflow.net/questions/13965 | -1 | (This is a follow-up question from over there: [Natural models of graphs](https://mathoverflow.net/questions/11647/natural-models-of-graphs).)
(And it *has* a follow-up question over there: [Naturally definable sets of natural numbers (2): Can the circle be broken?](https://mathoverflow.net/questions/14211/naturally-... | https://mathoverflow.net/users/2672 | Naturally definable sets of natural numbers |
>
> Question 3: Can every finite set of natural numbers be defined by a natural formula?
>
>
>
If I understand correctly, the answer is yes: We can always rewrite $x = n\_0 \vee x = n\_1 \vee ... \vee x = n\_k$ in the form $p(x) = q(x)$ for some polynomials $p$ and $q$ with natural number coefficients, which see... | 4 | https://mathoverflow.net/users/126667 | 14020 | 9,429 |
https://mathoverflow.net/questions/14024 | 8 | Algebraic geometry predicts a degree 2 branched cover from an elliptic curve to the projective line. What does this map look like topologically?
| https://mathoverflow.net/users/3238 | Degree 2 branched map from the torus to the sphere | One example: lay your $g$-holed torus $T$ out flat and draw a line the long way through each hole. It hits the torus in $2g + 2$ points. Consider the 180 degree rotation $w$ through that line. Now consider the space $T/w$ formed by identifying two points $P$ and $Q$ if $P = wQ$ (since $w^2 = 1$ we also have $Q = wP$). ... | 7 | https://mathoverflow.net/users/3384 | 14027 | 9,433 |
https://mathoverflow.net/questions/14012 | 3 | Let $k$ be a perfect field (so reduced = geometrically reduced) and $f:X\rightarrow \mathrm{Spec}(k)$ a Cohen-Macaulay morphism. Denote by $i:X\_{red}\rightarrow X$ the underlying reduced subscheme and by $\omega\_{X}$ and $\omega\_{X\_{red}}$ the relative dualizing sheaves of $X$ and $X\_{red}$ over $k$. What can one ... | https://mathoverflow.net/users/2215 | Adjunction for underlying reduced subschemes | Dear Bryden,
Hopefully I have things straight, and there is a general formula $i^!\omega\\_X = \omega\\_{X\\_{red}}$. One then has the functorial isomorphism (of sheaves on $X$)
$RHom\_{\mathcal O\_{X\\_{red}}}({\mathcal F},\omega\_{X\\_{red}}) = Rhom\_{\mathcal O\_X}(i\_\\*{\mathcal F}, \omega\\_X),$
for a coherent ... | 3 | https://mathoverflow.net/users/2874 | 14038 | 9,438 |
https://mathoverflow.net/questions/14022 | 4 | This is again a question asked to me by [this user](https://mathoverflow.net/users/3582/). He apparently quit using MO due to a busy time in personal and professional life and resulting difficulties in spending time here with patience. I am taking the liberty to ask it myself(with permission) as I consider him and his ... | https://mathoverflow.net/users/2938 | Characterizing nilpotents in a ring by a universal property | Let $e \in A$ be a non-zero idempotent (and hence not nilpotent).
Then if $f(e)$ is a unit, we find that $f(e) = 1,$
and so $f(e - 1) = 0.$ Thus if $e - 1$ generates (as a two-sided ideal) the entire
ring, we find that $f$ is identically zero, and hence that $B = 0$.
Thus, if we can find a non-zero idempotent $e \... | 8 | https://mathoverflow.net/users/2874 | 14041 | 9,440 |
https://mathoverflow.net/questions/14051 | 6 | This question was prompted by [the post here](https://mathoverflow.net/questions/14024/), and I asked this earlier, deleted it, and due to [pressure exerted by Ilya Nikokoshev](http://mathoverflow.tqft.net/discussion/194/), I am asking it again. Apologies to Pavel Etingof.
Q1. Let $\Lambda$ be a lattice in $\mathbb{C... | https://mathoverflow.net/users/2938 | Zeros of the Weierstrass $\wp$-function | There is an explicit formula for the zeroes: [Eichler, M.; Zagier, D. On the zeros of the Weierstrass $\wp$-function. Math. Ann. 258 (1981/82), no. 4, 399--407. [MR0650945 (83e:10031)](http://www.ams.org/mathscinet-getitem?mr=MR0650945)]
| 6 | https://mathoverflow.net/users/1409 | 14056 | 9,448 |
https://mathoverflow.net/questions/14058 | 11 | I'm interested in the structures of categories like $Rep(GL\_n), Rep(SL\_n)$, etc. of algebraic representations of an algebraic group. I understand that there should be some relation between these and the categories of representations of the corresponding Lie algebras.
However, it's not as intuitive to me what's going... | https://mathoverflow.net/users/344 | How do you switch between representations of an algebraic group and its Lie algebra? | If $G$ is semisimple simply connected in characteristic zero, the differential at $1$ gives an equivalence of (tensor) categories $Rep(G)\to Rep({\mathfrak g})$. If $G$ is not semisimple, this is not the case, but this functor is always fully faithful (i.e. an equivalence onto a full subcategory) if $G$ is connected. T... | 22 | https://mathoverflow.net/users/3696 | 14064 | 9,452 |
https://mathoverflow.net/questions/14054 | 25 | Does there exist anywhere a comprehensive list of small genus modular curves $X\_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly interested in genus one cases, and moreso in split/non-split cartan, with or without normalizers.
Ken Mcmurdy has a list [here](http://phob... | https://mathoverflow.net/users/2 | Where can I find a comprehensive list of equations for small genus modular curves? | No, there does not exist a comprehensive list of equations: the known equations are spread out over several papers, and some people (e.g. Noam Elkies, John Voight; and even me) know equations which have not been published anywhere.
When I have more time, I will give bibliographic data for some of the papers which giv... | 22 | https://mathoverflow.net/users/1149 | 14075 | 9,461 |
https://mathoverflow.net/questions/14076 | 77 | Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this [paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa82/aa8237.pdf) (Brindza and Pintér, *On the irreducibility of some polynomials in two variables*, Acta Arith. 1997).
Does anyone know of simila... | https://mathoverflow.net/users/2083 | Irreducibility of polynomials in two variables | A trick which works surprisingly often in my experience: If the Newton polytope of $f$ can not be written as a Minkowski sum of two smaller polytopes, then $f$ is irreducible. I think of this as a generalization of Eisenstein's criterion.
It is surprisingly easy to test whether a lattice polygon in $\mathbb{R}^2$ ca... | 100 | https://mathoverflow.net/users/297 | 14080 | 9,464 |
https://mathoverflow.net/questions/14093 | 13 | Let ${\cal M}\_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}\_g\subset {\cal M}\_g$ be the hyperelliptic locus and set ${{\cal H}}'\_g$ to be the preimage of ${\cal H}\_g$ in the Teichmueller space.
While working on a problem I arrive at two results that can't be reconciled unless ${\cal H}... | https://mathoverflow.net/users/2349 | Hyperelliptic loci in Teichmueller spaces | That $\mathcal H'\_g$ can't be connected for $g \geq 3$, isn't this just the "lift to Teichmuller space" of the result that the mapping class group isn't the hyperelliptic group?
In particular, the path components of $\mathcal H'\_g$ are indexed by the cosets of the hyperelliptic group in the mapping class group. No?... | 7 | https://mathoverflow.net/users/1465 | 14095 | 9,474 |
https://mathoverflow.net/questions/13817 | 37 | This is a followup to a previous question
[What is the right definition of the Picard group of a commutative ring?](https://mathoverflow.net/questions/13768/what-is-the-right-definition-of-the-picard-group-of-a-commutative-ring)
where I was worried about the distinction between invertible modules and rank one proj... | https://mathoverflow.net/users/1149 | A finitely generated, locally free module over a domain which is not projective? | It is impossible to produce an example of a finitely generated flat $R$-module that is not projective when $R$ is an integral domain. See: Cartier, "Questions de rationalité des diviseurs en géométrie algébrique," [here](http://www.numdam.org/numdam-bin/item?id=BSMF_1958__86__177_0), Appendice, Lemme 5, p. 249. Also se... | 44 | https://mathoverflow.net/users/3049 | 14111 | 9,479 |
https://mathoverflow.net/questions/13784 | 5 | Let $X\_t$ be an ergodic (time-homogeneous) Markov process (in discrete or continuous time) on a finite state space $\{1,\dots,n\}$. Let $T(X\_0)$ be the stopping time given by the infimum of times such that $X$ has covered the space (i.e., for all $j$ with $1 \le j \le n$ there exists some $t\_j \le T(X\_0)$ s.t. $X\_... | https://mathoverflow.net/users/1847 | Is there a way to analytically compute the recurrence time of a finite Markov process? | This is a response to a comment.
The [coupon collector's problem](http://en.wikipedia.org/wiki/Coupon_collector%27s_problem) is elementary. I don't have a particular scholarly reference in mind, but rather the technique of the proofs. There are a few proofs of the $n H\_n$ expected time to collect all coupons. One po... | 2 | https://mathoverflow.net/users/2954 | 14122 | 9,488 |
https://mathoverflow.net/questions/13972 | 4 | This question is about a puzzle from the book of Raymond Smullyan: The Lady or the Tiger? The description of the puzzle starts in Chapter 8, p. 103, and here is the important part (copy-pasted from Bruce Ediger's webpage):
Property Q: For any combination x, the combination QxQ is specially related to x.
Property L:... | https://mathoverflow.net/users/955 | Shortest Key for the Monte Carlo Lock of Smullyan | Edited in recognition of closed-mindedness.
~~My brute force search shows no keys shorter than 10. Here are the only keys of length 10 and 11 respectively:~~
RVLVQRVLVQ
VRLVQVRLVQ
VLRVQVLRVQQ
VLVRQVLVRQQ
Curiously, there are no keys of length 12.
The only word of length 7 that does not crash under iteration ... | 3 | https://mathoverflow.net/users/2368 | 14127 | 9,492 |
https://mathoverflow.net/questions/14133 | 12 | Is there a category (in the category theory sense) of non-well-founded sets (something analogous to Set, the category of sets), and has it been (well-)studied? Any references are appreciated.
| https://mathoverflow.net/users/1015 | Is there a category of non-well-founded sets? | Yes, there is. See Peter Aczel's 1988 notes on models of non-well-founded set theory, [here](http://standish.stanford.edu/pdf/00000056.pdf). The basic idea is that you can model sets as graphs (ie, as a collection of element sets together with a binary relation intuitively suggesting membership), and then the foundatio... | 14 | https://mathoverflow.net/users/1610 | 14135 | 9,497 |
https://mathoverflow.net/questions/13977 | 13 | Suppose that I have a nice variety *X* over ℚp, with good reduction if you like, and a nice sheaf on *X*, say coming from a smooth group scheme *G*. I can cover *X* by some p-adic open sets *Uα*, for example the mod-p neighbourhoods coming from some model $\mathcal{X}$ of *X*. Clearly I can't expect to use Čech cohomol... | https://mathoverflow.net/users/3753 | Is there something like Čech cohomology for p-adic varieties? | The first comment to make is that Cech theory is really extremely general, and can be set up to compute the cohomology of any complex of abelian sheaves on any site (provided you have coverings that are cohomologically trivial). This is explained at least somewhat in SGA4, Expose 5 and EGA III, Chap 0, section 12.
I... | 13 | https://mathoverflow.net/users/2215 | 14146 | 9,502 |
https://mathoverflow.net/questions/14151 | 7 | I would like to learn about derived functors.
Which reference do you advise ?
| https://mathoverflow.net/users/2330 | A good place where to learn about derived functors | I have to agree strongly with Ame's answer, in part. Weibel is a great place to go for the formalism. Once you have a little bit of the formalism, though, where to go depends on interests. To really see a lot of the power of derived functors, Hartshorne chapter 3 has some good theorems and exercises using them (though ... | 7 | https://mathoverflow.net/users/622 | 14155 | 9,509 |
https://mathoverflow.net/questions/14062 | 2 | How do I factorize a polynomial $X^n - 1$ over $\mathbb{F}\_p$? In particular I need to find factors of the polynomial $X^{3^3 - 1} - 1 = X^{26} - 1$ over $\mathbb{F}\_3$.
| https://mathoverflow.net/users/507 | How to factorize X^n - 1 in Z/pZ? | If you just need a quick answer (to decide if something else is going to work how you need), then you can do this with Wolfram|Alpha. Go there:
<http://www.wolframalpha.com/>
and input "factor x^26-1" and press the "equal" button. It'll show some info about the polynomial, including the factors mod 2. In many boxes, t... | 10 | https://mathoverflow.net/users/935 | 14156 | 9,510 |
https://mathoverflow.net/questions/14129 | 1 | For $A \in SU(2,C)$, it is clear that $A$ is completly determined by its first row (well any row or column, but let's say first column). In the general $SU(n,C)$-case this is no longer true. In fact, it seems that for every complex $n$-vector of unit norm, there exists a family of matrices for which the vector is the f... | https://mathoverflow.net/users/3787 | Describing $SU(n,C)$ | Use the fact that matrices act on vectors. $SU(n)$ acts transitively on the space of unit-length vectors; the stabilizer of a point is $SU(n-1)$ by Thorny's argument. For example, for the vector $(1,0,...,0)$ the stabilizer is the subgroup $\left(\begin{matrix}1&0\\\\ 0&A\end{matrix}\right)\approx SU(n-1)$. Now by the ... | 3 | https://mathoverflow.net/users/250 | 14169 | 9,520 |
https://mathoverflow.net/questions/14177 | 29 | We know that any smooth projective curve can be embedded (closed immersion) in $\mathbb{P}^3$. By definition a projective scheme over $k$ admits an embedding into some $\mathbb{P}^n$. Can we create an upper bound for the $n$ required (perhaps by strengthening the hypotheses) necessary to create an embedding of a smooth... | https://mathoverflow.net/users/3779 | Is there a Whitney theorem type theorem for projective schemes? | Over an algebraically closed field, any projective smooth variety of dimension $n$ can be embedded in $\mathbb P^{2n+1}$. This is elementary and can be found in Shafarevich's Basic Algebraic Geometry, Chapter II, §5.4 .
Of course specific varieties might be embedded in projective spaces of lower dimension.
For an abe... | 29 | https://mathoverflow.net/users/450 | 14183 | 9,529 |
https://mathoverflow.net/questions/14186 | 7 | Lemma 1 from [Anderson & Trapp's Shorted Operators, II](http://www.jstor.org/stable/2100462) is
> Let $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ran($B$).
(2) $AA^\* \le \lambda^2 BB^\*$ for some $\lambda \ge 0$.
(3) There ... | https://mathoverflow.net/users/238 | On operator ranges in Hilbert & Banach spaces | (1) does not generally imply (3) for bounded operators between Banach spaces. The first example I have a reference for was due to Douglas and was included in ["Factorization of operators on Banach space"](http://www.jstor.org/stable/2038955) by Embry in 1973. That paper has much more that might interest you, such as th... | 5 | https://mathoverflow.net/users/1119 | 14191 | 9,533 |
https://mathoverflow.net/questions/14195 | 0 | (All rings assumed to be commutative and unital)
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for $rad(\mathfrak{p}S)$ to be prime? How about when $S$ is integral over $R$? How about when it is ... | https://mathoverflow.net/users/1353 | When is the radical of the extension of a prime ideal prime? | Probably Harry has already thought of it like this, but rephrased in terms of algebraic geometry, this question is basically "under what conditions is the preimage of an irreducible variety under a map be irreducible."
When you phrase it like that, it sounds like you might be able to get a statement like "when all ma... | 4 | https://mathoverflow.net/users/66 | 14196 | 9,536 |
https://mathoverflow.net/questions/14206 | 4 | Sorry for not knowing the answers to these elementary questions:
Is the property of formulas of the first-order language of Peano arithmetic of "defining a finite set of natural numbers" goedelizable?
If so: Is the set of formulas with this property decidable, semidecidable or non-decidable?
| https://mathoverflow.net/users/2672 | Goedelizability and decidability of a property of Peano formulas | The set of (Gödel codes for) PA provably bounded formulas $\phi(x)$ is computably enumerable (c.e.). By provably bounded, I mean PA $\vdash \exists b\forall x(\phi(x)\to x \leq b)$. Indeed, you can enumerate all consequences of PA and when you find one of the shape $\exists b\forall x(\phi(x)\to x \leq b)$ then enumera... | 10 | https://mathoverflow.net/users/2000 | 14207 | 9,545 |
https://mathoverflow.net/questions/14214 | 9 | background/motivation
---------------------
let *Ek* denote the modular form of level one and weight *k* with *q*-expansion given by $E\_k(q)=1- \frac{2k}{b\_k}\sum\_n \sigma\_{k-1}(n)q^n$ where σ*i* is the [divisor sum](http://en.wikipedia.org/wiki/Divisor_function) and *bk* is the *k*-th [bernoulli number](http://e... | https://mathoverflow.net/users/2615 | how do you evaluate the p-adic modular form E_p-1 in the region |j|<1 | One has $j = E\_4^3/\Delta$. In the region $|j|\leq 1$, one is parameterizing elliptic curves with good reduction, and so $\Delta$ is a unit. Thus $|j| = |E\_4|^3$. This will help you
when $p = 5$.
When $p = 7,$ one can write $j = 1278 + E\_6^2/\Delta,$ hence $|E\_6|^2 = | j - 1728|$ on
the region $|j| \leq 1$.
F... | 5 | https://mathoverflow.net/users/2874 | 14220 | 9,552 |
https://mathoverflow.net/questions/14211 | -1 | (follow-up to: [Naturally definable sets of natural numbers](https://mathoverflow.net/questions/13965/naturally-definable-sets-of-natural-numbers))
Every formula $\Psi(x)$ in the first-order language of Peano arithmetic defines a set of natural numbers. Some of these sets are finite, others are infinite. Every finite... | https://mathoverflow.net/users/2672 | Naturally definable sets of natural numbers (2): Can the circle be broken? | From what I gather from your question, your "natural formulas" would form a complete set of distinct representatives for definable subsets of N under the equivalence relation $X \mathrel{E}\_0 Y$ defined by $|(X \setminus Y) \cup (Y \setminus X)| < \aleph\_0$. Unfortunately, there is no simple way to do this; such a sy... | 9 | https://mathoverflow.net/users/2000 | 14221 | 9,553 |
https://mathoverflow.net/questions/14210 | 2 | Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \to Spec(R)$ to be injective as a map of topological spaces (completely disregarding any of our sheaf structure)? That is,... | https://mathoverflow.net/users/1353 | When do primes lift uniquely (provided they lift at all)? | Rather than listing various non-trivial sufficient conditions, let me give you a reference. Search EGA for the word "radiciel" (or start reading at Definition 3.5.4 of EGA I). A morphism of schemes $X\to Y$ is said to be *radiciel* (or *universally injective*) if for every field $K$, the induced map on $K$-points $X(K)... | 2 | https://mathoverflow.net/users/1 | 14222 | 9,554 |
https://mathoverflow.net/questions/14007 | 5 | Hi,
Given a $\sigma$-unital $C^\*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon\_i\}, i\in I$ in $A$ such that each $\epsilon\_i$ is of the form $< e\_i,e\_i>\_A$, where $e\_i \in E, \forall i\in I$? If not, what are the conditions on $E$ and $A$ for which thi... | https://mathoverflow.net/users/3494 | Hilbert $C^*$-modules and approximate units | No, there is not always such an approximate unit. (This will be easier to formulate in terms of left modules, and with inner products linear in the first entry. The warning seems necessary due to the common convention in C\*-module theory to do the opposite.)
### Example
Let $A=B(\mathbb{C}^2)$ (linear operators), ... | 2 | https://mathoverflow.net/users/1119 | 14227 | 9,559 |
https://mathoverflow.net/questions/14219 | 41 | I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show that the special Schubert classes satisfy Pieri's formula. Then use the fact that basic homogeneous symmetric functions $h... | https://mathoverflow.net/users/321 | Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian? | There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just ~~Ehresmann's~~ Lesieur's proof as recounted in the OP).
The wrong one is $Rep(GL\_d)$, whose basis is indexed by decreasing sequences in ${\mathbb Z}^d$.
That has a subring $Rep(M\_d)$, rep... | 31 | https://mathoverflow.net/users/391 | 14231 | 9,563 |
https://mathoverflow.net/questions/13593 | 5 | Is it true that any monomorphism of commutative Hopf algebras over a field is injective? Moreover, is it true that any epimorphism of commutative Hopf algebras over a field is surjective?
| https://mathoverflow.net/users/3824 | Is every monomorphism of commutative Hopf algebras (over a field) injective? | It seems that Ben is nevertheless right that the answer to the first question is "NO". Let $G=SL(2,\Bbb C)$, and $B$ be the subgroup of lower triangular matrices. Then the inclusion $B\to G$ is an epi, since every algebraic representation of $B$ that extends to $G$ does so uniquely (on the nose, not just up to an isomo... | 2 | https://mathoverflow.net/users/3696 | 14234 | 9,566 |
https://mathoverflow.net/questions/14235 | 0 | By Heisenberg group I mean the group with presentation $H$ generated by $x$ and $y$ such that $x$ and $y$ commute with $xyx^{-1}y^{-1}$. Is there an infinite chain of subgroups
$H > H\_1 > H\_2 > \dots$ such that the index $[H\_i: H\_{i+1}]< n$ for some $n\ ?$ Thanks
| https://mathoverflow.net/users/3804 | decreasing chain of subgroups in the Heisenberg group | Assuming this *is* the discrete Heisenberg group $H=H\_3({\mathbb Z})$, as in my comment above, then here is another way of looking at Mariano's answer (I think). Take any sequence of positive integers
$n\_1 < n\_2 < \dots $ where $n\_i \vert n\_{i+1}$ for all $i$, and put
$$ H\_i = H\_3(n\_i{\mathbb Z}) $$
(Marian... | 4 | https://mathoverflow.net/users/763 | 14240 | 9,569 |
https://mathoverflow.net/questions/14257 | 5 | Forgive me for this naive question.
We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, [page 784](https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA784).
>
> Every module is a submodule of an injective module.
>
>
>
For a module $M$, Lang defin... | https://mathoverflow.net/users/2938 | Injective modules and Pontrjagin duals | Why this works? Because $\mathbb{Q}/\mathbb{Z}$ is an [injective cogenerator](https://en.wikipedia.org/wiki/Injective_cogenerator) in the category of abelian groups.
The mysterious appearance can be explained if you look at it from a more general perspective. This would be Morita duality, which was motivated by class... | 16 | https://mathoverflow.net/users/2384 | 14260 | 9,580 |
https://mathoverflow.net/questions/14264 | 0 | The integral I am interested in is:
$$t(x)=\int\_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\int\_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc has radius $K$
Is it correct that for K<1 this integral is:
$$-\int\... | https://mathoverflow.net/users/3589 | An integral arising in statistics(2) | For $K=1$ your arc passes through the pole of the function $\frac{exp(ixy)}{1+y^2} = \frac{exp(ixy)}{2i}\left(\frac{1}{y-i}-\frac{1}{y+i}\right)$, so you don't get a sensible value (the discontinuity of the integrand is asymptotically $\frac{c}{t}$ for $t$ around $0$).
| 1 | https://mathoverflow.net/users/2368 | 14265 | 9,582 |
https://mathoverflow.net/questions/14237 | 8 | Any finitely presented group naturally gives rise to an edge-labeled graph (the Cayley graph) and I am considering paths through this graph. Paths correspond to infinite sequences of generators, so we can ask questions about the recursiveness (or not) of these sequences\*. For example, is there a group where any recurs... | https://mathoverflow.net/users/3806 | How bad can the recursive properties of finitely presented groups be? | Your question is very interesting. I don't have a complete
answer.
First, let me note that in a finitely presented group, the
Cayley graph itself may not be a decidable graph, since to
know whether or not a node in the graph, which is a word in
the presentation, is trivial or not amounts exactly to the
word problem f... | 4 | https://mathoverflow.net/users/1946 | 14274 | 9,585 |
https://mathoverflow.net/questions/14251 | 6 | Introduction
------------
Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$
$$V\_{n,m}=\bigotimes\_{k=1}^n \mathbb C^m$$ and the symmetric group $S\_n$ acts on $V\_{n,m}$ by permutation of factors. Now the space $V\_{n,m}$ br... | https://mathoverflow.net/users/2384 | Asymptotics of symmetry types of tensors | This is not a complete answer, but perhaps will help. The probability distribution on tuples that shows up in your Theorem 1 is well known: It is the joint probability density function for the eigenvalues of a random unitary matrix, in the standard ``Gaussian unitary ensemble". See equation (9) in the paper of Terry Ta... | 4 | https://mathoverflow.net/users/1799 | 14295 | 9,596 |
https://mathoverflow.net/questions/14283 | 16 | As [noted earlier](https://mathoverflow.net/questions/14278/), I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and clean.
In any case, the second part of the book is "thoroughly unmodern", in... | https://mathoverflow.net/users/2938 | Central simple algebras approach to class field theory, merits of | If you are looking for a more conceptual way of understanding the central simple algebras approach to class field theory, I think that you would be well advised to look at Roquette's book [The Brauer-Hasse-Noether Theorem in Historical Perspective](http://books.google.com/books?id=ICnySc00PA4C&printsec=frontcover&dq=th... | 16 | https://mathoverflow.net/users/nan | 14297 | 9,597 |
https://mathoverflow.net/questions/14267 | 6 | If $U \subset \mathbb C^n$ is the complement of a closed analytic subset of codimension at least three then there is a result of Cartan which says that $H^1(U,\mathcal O^{analytic}\_U)=0$, see page 133 of "Theory of Stein Spaces" by Grauert and Remmert.
Does anyone know a reference for the analogous result in the alg... | https://mathoverflow.net/users/605 | Reference for cohomology vanishing | To expand on Emerton's [answer](https://mathoverflow.net/a/14270): Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at least $3$, then $H^i\_I(R)=0$ for $i\leq 2$.This follows because: $$H^i\_I(R) = \lim ... | 4 | https://mathoverflow.net/users/2083 | 14298 | 9,598 |
https://mathoverflow.net/questions/14300 | 1 | I have another question about $SU(n)$, again I hope it's not too basic. For $n=2$, the action of $SU(2)$ on $C^2$ is free since it's equal to the group of rotations. In general, the group of rotations is properly contained in $SU(n)$, does this mean that its action on $C^n$ is no longer free.
| https://mathoverflow.net/users/3787 | Freeness of the Canonical $SU(n)$ Action | $SU(n)$ acts freely on $\Bbb C^n\setminus 0$ (not on $\Bbb C^n$) for $n=1,2$, but not for higher $n$ (the stabilizer of a point is $SU(n-1)$). If this is what you are asking about, the answer is "yes". (This IS a basic question, though.)
| 1 | https://mathoverflow.net/users/3696 | 14302 | 9,601 |
https://mathoverflow.net/questions/13374 | 4 | Everybody knows that equality is reflexive: $\forall(x)(x=x)$. But should reflexivity of equality be taken as an axiom of logic or as a theorem of set theory?
If you choose the former then you probably need the axiom of extensionality: $\forall(x)\forall(y)(x=y\leftrightarrow\forall(z)(z\in x\leftrightarrow z\in y))$... | https://mathoverflow.net/users/3554 | Is reflexivity of equality an axiom or a theorem? | I managed to write down a proof for the reflexivity of equality using only the definition of equality in terms of membership and the rules of natural deduction.
1. Premise: $\forall x\_0\forall x\_1\left(\left(x\_0=x\_1\right)\leftrightarrow\forall x\_2\left(\left(x\_2\in x\_0\right)\leftrightarrow\left(x\_2\in x\_1\... | 1 | https://mathoverflow.net/users/3554 | 14304 | 9,602 |
https://mathoverflow.net/questions/14278 | 19 | I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it (thanks to [some comments](https://mathoverflow.net/questions/13106/map-of-number-theory/13134#13134) by Ben Linowitz).
Right from the very beginning, Weil uses the fact that when you have a locally compact t... | https://mathoverflow.net/users/2938 | Haar measure on a quotient, References for | The book I always look at for such things is Nachbin, The Haar Integral, which is short, and has a whole chapter on Integration on Locally Compact Homogeneous Spaces.
And a plus: he gives you a choice of reading the proof of the existence and uniqueness of the Haar integral according to Weil or according to Henri Ca... | 27 | https://mathoverflow.net/users/930 | 14305 | 9,603 |
https://mathoverflow.net/questions/14296 | 3 | Following on from this question [link text](https://mathoverflow.net/questions), how are the line bundles of a complex flag variety indexed? Are they still labeled by the integers? If so, why? A representation theory explanition in terms of the homogenous space description of the variety $U(n)/U(k\_1) \times \cdots \ti... | https://mathoverflow.net/users/1648 | Indexing the Line Bundles of a Flag Manifold | Attempting to answer your last question: given a (topological, Lie, algebraic, etc.) group G and a closed subgroup P in G, the category of G-equivariant vector bundles on X=G/P is equivalent to the category of representations of P. So to a representation of P one can assign a vector bundle over G/P, and even a G-equiva... | 5 | https://mathoverflow.net/users/2106 | 14308 | 9,606 |
https://mathoverflow.net/questions/14266 | 23 | I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do homological algebra for commutative monoids, but first let me explain some background, my motivation, and articulate mor... | https://mathoverflow.net/users/184 | A Peculiar Model Structure on Simplicial Sets? | So the short answer is that there is not such a model structure. The difficulty arises in trying to show that the class of weak equivalences has all of the necessary properties; in particular, even two-of-three does not hold for the naive definition. The first difficulty arises even before that: on ordinary simplicial ... | 12 | https://mathoverflow.net/users/1378 | 14310 | 9,608 |
https://mathoverflow.net/questions/14311 | 27 | I was reading [this question](https://mathoverflow.net/questions/731/points-in-algebraic-geometry-why-shift-from-m-spec-to-spec) on why algebraic geometry looks at prime ideals instead of only maximal ideals, and I understand Anton's answer, but I'm a little confused as to how this fits with Hilbert's Nullstellensatz -... | https://mathoverflow.net/users/1916 | MaxSpec, Spec, ... "RadSpec"? Or, why not look at all radical ideals? | The short answer is that every radical ideal is the intersection of the prime ideals containing it, so that the pullback map on Specs determines the pullback map on your wouldbe RadSpecs.
Note that a similar reason underlies the success of looking only at maximal ideals in classical algebraic geometry: a finitely ge... | 25 | https://mathoverflow.net/users/1149 | 14312 | 9,609 |
https://mathoverflow.net/questions/14314 | 51 | The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
Question: Are there other categories of algebraic objects that have interesting natural topologies that carry algebraic ... | https://mathoverflow.net/users/1353 | "Algebraic" topologies like the Zariski topology? | Yes, there are plenty of such things.
[In the following, "compact" implies "locally compact" implies "Hausdorff".]
1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.
(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zaris... | 50 | https://mathoverflow.net/users/1149 | 14315 | 9,611 |
https://mathoverflow.net/questions/14273 | 16 | When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of complex $\zeta$-functions and $L$-functions. As a preliminary list, we fix the following list. But feel free to add to it.
... | https://mathoverflow.net/users/2938 | PNT for general zeta functions, Applications of. | Hi Anweshi,
Since Emerton answered your third grey-boxed question very nicely, let me try at the first two. Suppose $L(s,f)$ is one of the L-functions that you listed (including the first two, which we might as well call L-functions too). (For simplicity we always normalize so the functional equation is induced by $s... | 9 | https://mathoverflow.net/users/1464 | 14318 | 9,614 |
https://mathoverflow.net/questions/14341 | 50 | [A recent answer](https://mathoverflow.net/questions/14311/maxspec-spec-radspec-or-why-not-look-at-all-radical-ideals/14319#14319) motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - latti... | https://mathoverflow.net/users/1916 | Good lattice theory books? | A good, user-friendly, modern, introductory textbook is Davey and Priestley's *Introduction to Lattices and Order*.
Incidentally, Gian-Carlo Rota used to say much the same thing as you, Zev: that lattice theory had been robbed of its rightful place in mathematics.
| 34 | https://mathoverflow.net/users/586 | 14342 | 9,628 |
https://mathoverflow.net/questions/14324 | 9 | Warning: I'll be using the "pre-$\lambda$-ring" and "$\lambda$-ring" nomenclature, as opposed to the "$\lambda$-ring" and "special $\lambda$-ring" one (although I just used the latter a few days ago on MO). It's mainly because both sources use it, and I am (by reading them) slowly getting used to it.
Let $G$ be a fin... | https://mathoverflow.net/users/2530 | Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113 | I've just gone and looked up [158] (Gay, C. D.; Morris, G. C.; Morris, I.
Computing Adams operations on the Burnside ring of a finite group.
J. Reine Angew. Math. 341 (1983), 87--97.
On p. 90, at the end of section 2, they say: "Knutson conjectured that the Adams operations on SCF(G) inherited from A(G) [=Burnside ri... | 6 | https://mathoverflow.net/users/437 | 14351 | 9,636 |
https://mathoverflow.net/questions/14338 | 38 | I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this.
From what I've seen, model theorists and logicians are mostly opposed to GCH, while on the other end of the spectrum, some functional analysis depends on GCH, so it is much... | https://mathoverflow.net/users/1353 | What is the general opinion on the Generalized Continuum Hypothesis? | There is definitely a not-CH tendency among set theorists with a strong Platonist bent, and my impression is that this is the most common view. Many of these set theorists believe that the large cardinal hierarchy and the accompanying uniformization consequences are pointing us towards the final, true set theory, and t... | 46 | https://mathoverflow.net/users/1946 | 14353 | 9,638 |
https://mathoverflow.net/questions/14358 | 6 | If one views a group as a one object category with the elements of the group as morphisms then a natural transformation between functors of such categories is an inner automorphism, i.e. if we have two group homomorphisms $f,g: A\to B$ then a natural transformation $\eta :f\to g$ is just an element $b\in B$ such that $... | https://mathoverflow.net/users/nan | groups as categories and their natural transformations | The comments thread is getting a bit long, so here's an answer. The category $C(G)$ that David associates to a group $G$ (by his second recipe) has the elements of $G$ as its objects, and exactly one morphism between any given pair of objects. It's what category theorists call an indiscrete or codiscrete category, and ... | 11 | https://mathoverflow.net/users/586 | 14364 | 9,644 |
https://mathoverflow.net/questions/14361 | 11 | In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined as summands of the pushforwards of the constant sheaves on stacks of quiver representations along with a choice of invar... | https://mathoverflow.net/users/66 | What do the local systems in Lusztig's perverse sheaves on quiver varieties look like? | For affine quivers, except cyclic ones, there are always perverse sheaves attached to nontrivial local systems.
If you just need an example, I recommend you to read McGerty's paper [math/0403279](http://arxiv.org/abs/math/0403279), before Lusztig's paper, where the Kronecker quiver case is studied in detail.
I also... | 10 | https://mathoverflow.net/users/3837 | 14366 | 9,646 |
https://mathoverflow.net/questions/14118 | 7 | Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y\_E(p)$, which sends L to set of elliptic curves F/L, and symplectic isomorphisms $\phi:E[p] \rightarrow F[p]$. We know that this moduli problem is representable by a curve o... | https://mathoverflow.net/users/92 | Cyclic extensions coming from E[p] \equiv F[p], | The answer is that in fact this construction does not produce cyclic extensions! The problem is that $X\_E(p^2) \to X\_E(p)$ is not generically Galois; it is so only after extension of the ground field.
Here is a more detailed explanation. Assume that $p \ge 3$ and that $p \nmid \operatorname{char} K$. Then $Y\_E(p^2... | 9 | https://mathoverflow.net/users/2757 | 14367 | 9,647 |
https://mathoverflow.net/questions/14238 | 16 | In question #7656, Peter Arndt asked [why the Gamma function completes the Riemann zeta function](https://mathoverflow.net/questions/7656/why-does-the-gamma-function-complete-the-riemann-zeta-function) in the sense that it makes the functional equation easy to write down. Several of the answers were from the perspectiv... | https://mathoverflow.net/users/290 | Why does the Gamma function satisfy a functional equation? | Your questions are a part of what Deninger has been writing about for 20 years. He's proposed a point of view that sort of explains a lot of things about zeta functions. It's important to say that this explanation is more in a theoretical physics way than in a mathematical way, in that, as I understand it, he's predict... | 13 | https://mathoverflow.net/users/1114 | 14373 | 9,652 |
https://mathoverflow.net/questions/14374 | 7 | By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book *Euler through time. A new look at old themes*, 2006) I found that $$\sum B\_n = \int\_0^\infty \frac{t}{e^{2t}-e^t} dt$$ and discovered numerically that this expression equals $\zeta(2)-1$. The... | https://mathoverflow.net/users/3503 | Integral expression for zeta(2) | The starting point is the integral
$$
\Gamma(s) = \int\_{0}^{\infty}e^{-x}x^{s-1}dx
$$
for the gamma function. Make the change of variable $x = nu$ with $n$ an arbitrary positive integer. Then
$$
\Gamma(s)n^{-s} = \int\_{0}^{\infty}e^{-nu}u^{s-1}du
$$
and summing over $n$ from $n = 1$ yields
$$
\Gamma(s)\ze... | 13 | https://mathoverflow.net/users/3304 | 14381 | 9,657 |
https://mathoverflow.net/questions/14371 | 19 | I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry texts considering the level of undergraduate students who have not learnt commutative algebra or homological algebra; the... | https://mathoverflow.net/users/1851 | Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course | [An invitation to algebraic geometry](http://www.amazon.fr/Invitation-Algebraic-Geometry-Karen-Smith/dp/0387989803) by Karen Smith is excellent; it is very intuitive, and does everything over the complex numbers. For absolute newcomers, this is probably the best introduction.
[Algebraic curves](http://www.math.lsa.um... | 28 | https://mathoverflow.net/users/1107 | 14382 | 9,658 |
https://mathoverflow.net/questions/14350 | 11 | Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a [reflexive](https://mathoverflow.net/questions/7490/differences-between-reflexives-and-projectives-modules/7588#7588) $R$-module and let $A=Hom\_R(M,M)$. Suppose $A$ has [finite global dimension](https://en.wikipedia.org/wiki/Global_dimension). Then ... | https://mathoverflow.net/users/2083 | Existence of non-commutative desingularizations | There are already counter-examples in dimension 2. If you take a 2-dml non-rational singularity, then there cannot exist a non-commutative resolution in your sense. In fact, any 2-dml nc-resolution your sense is also an nc-resolution in Michel's sense and so must have rational singularities by a [result of Toby Staffor... | 8 | https://mathoverflow.net/users/2275 | 14389 | 9,663 |
https://mathoverflow.net/questions/14388 | 10 | Jack symmetric polynomials are known to be generalizations of Schur functions $\chi\_\lambda$, for which powerful Weyl determinant formulas are known.
Are there any generalizations of two determinant formulas for general Jack symmetric $P^\alpha\_\lambda(x)$ functions?
The first determinant (Jacobi-Trudi) formula r... | https://mathoverflow.net/users/3840 | Jack polynomials as determinants | A natural analog would be a representation of the Jack polynomial as an alternating sum of eigenfunctions of the Calogero-Sutherland operator over an orbit of the symmetric group. This can be done even in the q-deformed case (i.e. for Macdonald polynomials and Macdonald operators). Such formula was conjectured by Felde... | 18 | https://mathoverflow.net/users/3696 | 14393 | 9,666 |
https://mathoverflow.net/questions/14404 | 53 | Has somebody translated J.-P. Serre's "Faisceaux algébriques cohérents" into English? At least part of it?
In a fit of enthusiasm, I started translating it and started TeXing. But after section 8, I got tired and stopped.
However if somebody else already took the trouble, I would be most grateful. I do not know a w... | https://mathoverflow.net/users/2938 | Serre's FAC in English | Together with some help from my friend, I translated FAC into English. I didn't have so much time to proofread it, so probably there are some mistakes.
It can be found here: [FAC](http://achinger.impan.pl/fac/fac.pdf), [Source](http://achinger.impan.pl/fac/fac.tar.gz).
| 89 | https://mathoverflow.net/users/3847 | 14417 | 9,683 |
https://mathoverflow.net/questions/14423 | 18 | I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is $G$ with Lie algebra $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm having a hard time, and I've realized this is because I don't have a good conceptual understanding of the Lie bracket, eit... | https://mathoverflow.net/users/492 | Understanding moment maps and Lie brackets | This question is (at least as I read it) about the Poisson bracket; the Poisson bracket is a Lie bracket structure on the functions on a symplectic manifold.
So how should one think about Poisson bracket? Well, remember that for every function on a symplectic manifold, one has a Hamiltonian vector field $X\_f$. One ... | 15 | https://mathoverflow.net/users/66 | 14427 | 9,690 |
https://mathoverflow.net/questions/14412 | 1 | Hello,
While reading the article Matrix Factorization Techniques for Recommender Systems I came across the following description:
"Matrix factorization models map both users and items to a joint latent factor space of dimensionality f, such that user-item interactions are modeled as inner products in that space."
... | https://mathoverflow.net/users/3845 | Matrix Factorization Model | I'll give an example from politics. Let's say you have a legislative body, such as the House of Representatives in the U. S. Congress. Over a period of time, the members of the House will vote on many bills and thereby accrue a voting history. Let's encode votes as numbers: a vote for a bill is 1, a vote against a bill... | 6 | https://mathoverflow.net/users/302 | 14437 | 9,697 |
https://mathoverflow.net/questions/10512 | 45 | [I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community. I have also voted to reopen it. -- PLC]
There are many ways to approach noncommutative geometry.
What are some of the most important currently known approaches? Who are the pr... | https://mathoverflow.net/users/2938 | Theories of Noncommutative Geometry | In accordance with the suggestion of Yemon Choi, I am going to suggest some further delineation of the approaches to "Non-commutative Algebraic Geometry". I know very little about "Non-commutative Differential Geometry", or what often falls under the heading "à la Connes". This will be completely underrepresented in th... | 28 | https://mathoverflow.net/users/348 | 14443 | 9,702 |
https://mathoverflow.net/questions/14450 | 11 | Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\mathcal S\mathfrak g)^{\mathfrak g}$ to be the ring of invariant elements of the symmetric algebra $\mathcal S\mathfrak g$ ... | https://mathoverflow.net/users/78 | Is there a canonical Hopf structure on the center of a universal enveloping algebra? | In the semisimple case, one has the Harish-Chandra isomorphism between the center ${\mathcal Z}(\mathfrak g)$ and $(S{\mathfrak h})^W$, where ${\mathfrak h}$ is a Cartan subalgebra of ${\mathfrak g}$ and $W$ is the Weyl group. On $S{\mathfrak h}$, there is a natural inner product $(f,g)=f(\partial)g(x)|\_{x=0}$ induced... | 8 | https://mathoverflow.net/users/3696 | 14457 | 9,708 |
https://mathoverflow.net/questions/14245 | 3 | Consider the optimization problem
$$\min\_x ||Ax||\_1 + \lambda||x-b||^2,$$
where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is closely related to the "lasso" problem in basis pursuit.) Can anything be said about the value of $\lambda$ for which $... | https://mathoverflow.net/users/1557 | Maximizing Sparsity in l1 Minimization? | The ultimate sparseness occurs when $Ax^\*(\lambda)=0$, which is the case when the minimizer $x^\*$ is the projection of $b$ onto $\ker A$. For this to happen, $\lambda$ must be small enough so that the restriction of $A$ to the orthogonal complement of its kernel is bounded from below by a constant greater than $2\lam... | 1 | https://mathoverflow.net/users/2912 | 14461 | 9,710 |
https://mathoverflow.net/questions/14446 | 8 | Let $C$ be a smooth curve of degree $d$ in $\mathbb{P}^2$ over $\mathbb{C}$. Say $C$ is defined by $p(x,y,z)=0$, with $p$ a homogeneous degree $d$ polynomial.
In vector calculus one learns that the gradient of $p$ is normal to $C$ at every point of the curve.
In algebraic geometry, the invertible sheaf associated t... | https://mathoverflow.net/users/1724 | Normal bundle to a curve in P^2 | Yes, there is a strong relationship between the two.
First, let's work locally in affine space rather than in projective space (it makes more
sense to work locally just because we are dealing with a sheaf, which is defined locally).
So I will consider a non-homogen
Working without a metric (as one does in at least ... | 9 | https://mathoverflow.net/users/2874 | 14470 | 9,716 |
https://mathoverflow.net/questions/14456 | 18 | Lagrange proved that every positive integer is a sum of 4 squares.
Are there general results like this for rings of integers of number fields? Is this class field theory?
Explicitly, suppose a number field is formally real. Denote its ring of integers by $Z$. Is it true that for every algebraic integer $x$ in $Z$... | https://mathoverflow.net/users/nan | sum of squares in ring of integers | To address the particularities of this question for number fields, the basic theorem is attributed to Hilbert, Landau and Siegel. First of all, any nonzero sum of squares in a number field has to be totally positive (that is, it is positive in all real embeddings). Hilbert (1902) conjectured that in any number field, a... | 51 | https://mathoverflow.net/users/3272 | 14473 | 9,719 |
https://mathoverflow.net/questions/14481 | -2 | Hi,
Could someone explain to me or point out some documentation on how to compute a given percentile from a histogram ?
| https://mathoverflow.net/users/3867 | calculate percentiles from a histogram | A histogram gives you the number $n\_i$ of observations between some $x\_i$ and $x\_{i+1}$. I'm assuming a total of $n$ observation. So, to get an approximation for the upper $p$-percentile, you want to find the maximal $j$ such that $\sum\_{i=j}^\infty n\_i\geq p\*n$. Then, the empirical upper $p$-percentile is betwee... | -1 | https://mathoverflow.net/users/2933 | 14482 | 9,724 |
https://mathoverflow.net/questions/13873 | 6 | The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does not generalize). I will give some background, and then ask my question as a conjecture, set apart from the main text.
... | https://mathoverflow.net/users/78 | Do there exist small neighborhoods in a classical mechanical system without pairs of focal points? | For a sufficiently large particle energy, the original problem can be transformed to a problem of geodesic motion as follows:
The motion of a classical particle in an external magnetic field in n-dimensions can be
seen as a symplectic reduction of a geodesic motion in n+1 dimensions (Rn \* S1) through the
Kaluza-Kl... | 3 | https://mathoverflow.net/users/1059 | 14494 | 9,730 |
https://mathoverflow.net/questions/14484 | 6 | I was looking through my notes for a homotopy theory course and found the following mysterious statement (*K* is of course the Eilenberg-Maclane space):
$$H^{n+1}(K(\mathbb Z\_p,n);\mathbb Z\_p) \cong \mathbb Z\_p.$$
(This would be obvious if *n+1* were replaced with *n*. This is supposed to imply that the natural... | https://mathoverflow.net/users/2467 | The (n+1)-st cohomology of K(Z/p,n). | Universal coefficient theorem + $H\_{n+1}(K(Z\_p,n);Z)=0$. An elementary way to see the latter is that the single $n+1$ cell added to $S^n$ to kill $p$ times the generator is not a cellular cycle.
| 12 | https://mathoverflow.net/users/3874 | 14497 | 9,732 |
https://mathoverflow.net/questions/14514 | 5 | Let G be a nontrivial finite group. Does there exist an irreducible representation of G of dimension greater than or equal to the cardinality of G?
[Edited for clarity. -- PLC]
| https://mathoverflow.net/users/3876 | How big can the irreps of a finite group be (over an arbitrary field)? | **EDIT:** Part 4 added. **EDIT2:** Second proof of Part 4 added.
**1.** The answer is no (as long as we are working over a field - of any characteristic, algebraically closed or not). If $k$ is a field and $G$ is a finite group, then the dimension of any irreducible representation $V$ of $G$ over $k$ is $\leq \left|G... | 28 | https://mathoverflow.net/users/2530 | 14516 | 9,740 |
https://mathoverflow.net/questions/14293 | 3 | This is a follow-up question to [my previous one](https://mathoverflow.net/questions/951) where I was trying to understand the classes of Legendrian immersions of circles into [contact manifolds](http://en.wikipedia.org/wiki/Contact_geometry).
I'm interested in classifying *isotropic* immersions (of spheres, in my pa... | https://mathoverflow.net/users/353 | Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds | jc, you'll have fun working out answers to examples of your first question. I'm only going to address the Legendrian case. If I didn't make mistakes, I'll conclude that Legendrian immersions of $S^n$ into $S^{2n+1}$ are all regular homotopic if $n$ is even, while if $n$ is odd they are classified by a rotation *number*... | 3 | https://mathoverflow.net/users/2356 | 14519 | 9,742 |
https://mathoverflow.net/questions/14508 | 18 | Suppose that $f$ is a weight $k$ newform for $\Gamma\_1(N)$ with attached $p$-adic Galois representation $\rho\_f$. Denote by $\rho\_{f,p}$ the restriction of $\rho\_f$ to a decomposition group at $p$. When is $\rho\_{f,p}$ semistable (as a representation of
$\mathrm{Gal}(\overline{\mathbf{Q}}\_p/\mathbf{Q}\_p)$?
To... | https://mathoverflow.net/users/2215 | Galois representations attached to newforms | The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D\_{pst}(\rho\_{f,p})$ by forgetting the filtration is precisely the one attached to $\pi\_p$, the representation of $GL\_2(\mathbf{Q}\_p)$ attached... | 15 | https://mathoverflow.net/users/1384 | 14521 | 9,744 |
https://mathoverflow.net/questions/14509 | 6 | Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullste... | https://mathoverflow.net/users/1916 | Various concepts of "closure" or "completion" in mathematics | In general, if $A\subset B$ is a full reflective subcategory, then each object $a\in A$ is isomorphic to its image under the reflector. This seems to include many cases: $\mathbf{Ab}\subset \mathbf{Grp}$, $\mathbf{CompMet}\subset\mathbf{Met}$, $\mathbf{Top}\_{n+1}\subseteq \mathbf{Top}\_{n}$ (as in [Why is Top\_4 a ref... | 7 | https://mathoverflow.net/users/2734 | 14533 | 9,753 |
https://mathoverflow.net/questions/14505 | 0 | An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4.
My question is: if the link bounds a surface with zero genus in B^4, is it necessarily a slice link? If not, any counter examples?
| https://mathoverflow.net/users/1537 | If the 4-genus of a link is zero, is it a slice link? | The Hopf link bounds a cylinder $S^1 \times [0,1]$ in $B^4$, and it's not slice since the two components have a non-zero linking number.
| 5 | https://mathoverflow.net/users/1465 | 14537 | 9,757 |
https://mathoverflow.net/questions/14529 | 18 | The Riemann–Stieltjes integral $\int\_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–Stieltjes sums are analogous to the Riemann sums $\sum\_{i=0}^{n-1} f(c\_i)(g(x\_{i+1})-g(x\_i))$ where $c\_i$ is in the $... | https://mathoverflow.net/users/1047 | Visualization of Riemann–Stieltjes Integrals | It's fairly easy to visualize the Riemann–Stieltjes integral $\int\_a^b f(t)\,dg(t)$ [I changed the name of the integration variable for convenience below] if $f\ge0$ and $g$ is nondecreasing. Just draw the graph of the curve $(x,y)=(g(t),f(t))$. The integral is just the area below the curve. (Whenever $g$ makes a jump... | 34 | https://mathoverflow.net/users/802 | 14539 | 9,759 |
https://mathoverflow.net/questions/14518 | 35 | This is related to Anweshi's question about [theories of noncommutative geometry](https://mathoverflow.net/questions/10512/).
Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative rings I have much truck with are either supercommutative, almost commutative (filtered, with c... | https://mathoverflow.net/users/622 | Applications of noncommutative geometry | Charles,
a couple of reasons why a complex algebraic geometer (certainly someone who is
interested in moduli spaces of vector bundles, as your profile tells me) might
at least keep
an open verdict on the stuff NC-algebraic geometers (NCAGers from now on) are trying to do.
in recent years ,a lot of progress has be... | 26 | https://mathoverflow.net/users/2275 | 14541 | 9,761 |
https://mathoverflow.net/questions/14530 | 4 | How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that?
Especially, any good idea how to calculate the 4-genus of a 2-bridge link? Thanks.
| https://mathoverflow.net/users/1537 | 4-genus of a 2-bridge link | You might get interesting lower bounds using [Rasmussen's s-invariant](http://arxiv.org/abs/math.GT/0402131), and his calculations of [the KR homology of 2-bridge links](http://arxiv.org/abs/math.GT/0508510).
Or you might not. As Ryan says, there are various ways of bounding genus above and below, but unless you get... | 5 | https://mathoverflow.net/users/66 | 14544 | 9,764 |
https://mathoverflow.net/questions/14501 | 9 | The background for this question is that I know that many students starting to learn algebra focus on the standard construction of the quotient group $G/N$ instead of working with the universal property. Therefore it would be nice to give another construction, thus avoiding cosets, but so that the universal property is... | https://mathoverflow.net/users/2841 | alternative construction of the quotient group | In order to make the construction of the quotient, it seems that it would be helpful to have at hand some quotients of the group $G$, described in some way other than formally via cosets.
Here is one possible approach:
Generalizing the regular action of $G$ on itself, we have the action of $G$ on its
power set (giv... | 3 | https://mathoverflow.net/users/2874 | 14554 | 9,772 |
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