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https://mathoverflow.net/questions/10959 | 4 | I recently heard the following puzzle: There are three nails in the wall, and you want to hang a picture by wrapping a wire attached to the picture around the nails so that if any one nail is removed the picture still stays but if any two nails are removed then the picture falls down. An answer, schematically, is given... | https://mathoverflow.net/users/303 | relationship between borromean rings and hanging-a-picture-from-three-nails puzzle? | This question is related to this post and its answers:
[Collapsible group words](https://mathoverflow.net/questions/15316/collapsible-group-words)
You may think of the three nails as giving a 3-punctured
plane, which has fundamental group a rank 3 free group.
An element of this group may be thought of as pushing a p... | 6 | https://mathoverflow.net/users/1345 | 18523 | 12,347 |
https://mathoverflow.net/questions/18514 | 5 | Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should be triangulated.
1. Is this possible in full generality? (Unbounded complexes, no restrictions on X)
2. Consider a sh... | https://mathoverflow.net/users/1261 | Closed monoidal structure on the derived category of sheaves | I think the natural context to answer this and related questions is that of symmetric monoidal stable $(\infty,1)$-categories. In that context Lurie's DAG I-III give all the necessary foundations to straightforwardly generalize all of the usual abelian story you refer to to the derived category. There is such an object... | 7 | https://mathoverflow.net/users/582 | 18524 | 12,348 |
https://mathoverflow.net/questions/18522 | 26 | So I did some algebraic topology at university, including homotopy theory and basic simplicial homology, as well as some differential geometry; and now I'm coming back to the subject for fun via Hatcher's textbook. A problem I had in the past and still have now is how to understand projective space RP^n - I just can't ... | https://mathoverflow.net/users/1256 | How should I visualise RP^n? | You can "visualize" the cell structure on $\mathbb{R}P^n$ rather explicitly as follows. The set of tuples $(x\_0, ... x\_n) \in \mathbb{R}^{n+1}$, not all equal to zero, under the equivalence relation where we identify two tuples that differ by multiplication by a nonzero real number, can be broken up into pieces depen... | 19 | https://mathoverflow.net/users/290 | 18530 | 12,352 |
https://mathoverflow.net/questions/18464 | 5 | Definitions and the main question
---------------------------------
Recall that a category $\mathcal C$ is **monoidal** if it is equipped with the following data (two functors, three natural transformations, and some properties):
* a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$,
* a functor $1: \{... | https://mathoverflow.net/users/78 | What are the correct axioms for a "weakly associative monoidal functor"? | Theo,
I believe the notion that you're looking for is called quasi-fiber functor, and is discussed in:
<http://www-math.mit.edu/~etingof/tenscat1.pdf>
It is also discussed how to do Tannakian formalism in the presence of a quass-fiber functor, and you get back a quasi-Hopf algebra, as you desire. I'll omit repeat... | 3 | https://mathoverflow.net/users/1040 | 18536 | 12,355 |
https://mathoverflow.net/questions/12068 | 17 | Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S\_n$). There is a well known, cool formula computing the Euler characteristic of all these n-symmetrical products:
$$\sum\_{d \geq 0} \ \chi \left(X^{[d]} \right)q^d... | https://mathoverflow.net/users/3314 | What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve? | For singular plane curves, there is a conjectural formula (due to Alexei Oblomkov and myself) in terms of the HOMFLY polynomial of the links of the singularities. For curves whose singularities are torus knots, i.e. like x^a = y^b for a,b relatively prime, and for a few more singularities, the conjecture has been estab... | 16 | https://mathoverflow.net/users/4707 | 18540 | 12,357 |
https://mathoverflow.net/questions/18544 | 13 | Is there a good way to define a sheaf over a simplicial set - i.e. as a functor from the diagram of the simplicial set to wherever the sheaf takes its values - in a way that while defined on simplex by simplex corresponds in some natural manner to what a sheaf over the geometric realization of the simplicial set would ... | https://mathoverflow.net/users/102 | Sheaves over simplicial sets | Clearly looking at sheaves on the geometric realisation gives something too far
removed from the simplicial picture. This is essentially because there are too
many sheaves on a simplex have (most of which are unrelated to simplicial
ideas). What one could do is to consider such sheaves which are constructible
with resp... | 10 | https://mathoverflow.net/users/4008 | 18549 | 12,361 |
https://mathoverflow.net/questions/18547 | 18 | I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it might also be interesting to ask what number of NxN (0,1)-matrices are singular or non-singular, I'd like to ignore singula... | https://mathoverflow.net/users/3737 | Number of unique determinants for an NxN (0,1)-matrix | I have given some detail in a comment to another answer. I have a proof that the number of determinants is greater than 4 times the nth Fibonacci number for (n+1)x(n+1) (0,1) matrices, and I conjecture that for large n the number of distinct determinants approaches a constant times n^(n/2). Math Overflow has some hints... | 4 | https://mathoverflow.net/users/3206 | 18554 | 12,364 |
https://mathoverflow.net/questions/18553 | 5 | Let $S\_4 = \left(\begin{array}{cc}0&-1 \\\ 1&0 \end{array}\right) \textrm{ and } S\_6 = \left(\begin{array}{cc} 1&-1 \\\ 1&0\end{array}\right)$. Serre proves in his book on trees that $SL\_2(\mathbb{Z}) \cong \mathbb{Z}/4 \*\_{\mathbb{Z}/2} \mathbb{Z}/6$, and $S\_4$ and $S\_6$ are the elements corresponding to the gen... | https://mathoverflow.net/users/2669 | Relations between two particular elements of SL_2(Z)? | Certainly $\mathrm{SL}(2,\mathbb{Z})$ contains a free group.
For instance $\Gamma(2)$, the subgroup of all matrices congruent
to the identity modulo $2$, is free of rank $2$. The matrices
$\left(\begin{array}{cc}1&2\\\ 0&1\end{array}\right)$
and
$\left(\begin{array}{cc}1&0\\\ 2&1\end{array}\right)$
freely generate $\Ga... | 11 | https://mathoverflow.net/users/4213 | 18556 | 12,366 |
https://mathoverflow.net/questions/18539 | 19 | As I understand it, [Lions and DiPerna demonstrated existence and uniqueness](http://www.jstor.org/pss/1971423) for the [Boltzmann equation](http://en.wikipedia.org/wiki/Boltzmann_equation). Moreover, [this paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VJ2-44D2CY3-J&_user=10&_coverDate=11%252F30%252... | https://mathoverflow.net/users/1847 | Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes? | Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at large in your question. I hope I can address at least some of them.
The "big picture" problem you are implicitly get... | 22 | https://mathoverflow.net/users/3948 | 18567 | 12,371 |
https://mathoverflow.net/questions/18551 | 4 | It follows from a [recent answer](https://mathoverflow.net/questions/16132/formally-etale-at-all-primes-does-not-imply-formally-etale/17775#17775) that even when a ring is formally étale rather than étale, we can check this condition on localizations, and hence stalks. It's not hard to show that we can define a "formal... | https://mathoverflow.net/users/1353 | Why do we need finiteness conditions for formally étale morphisms? | I guess, to get a good theory, you would want to be able to have effective descent for certain classes maps. In Brian Conrads answer to this question:
"[Quasi-separatedness for Algebraic Spaces](https://mathoverflow.net/questions/16381/quasi-separatedness-for-algebraic-spaces)"
he gives references to how you can wo... | 2 | https://mathoverflow.net/users/1084 | 18587 | 12,385 |
https://mathoverflow.net/questions/18429 | 9 | Following the first chapter of Hatcher's great book "Spectral Sequences in Algebraic Topology", I got into problems with spectral sequences of cohomological type. Fix a ring $R$ once and for all. Please let me first recapitulate the **homological situation**.
An *exact couple* consists of bigraded $R$-modules $A$ and... | https://mathoverflow.net/users/4676 | Convergence of spectral sequences of cohomological type | The lemma you refer to has two halves. The first half covers the case of homology and the second half covers the case of cohomology. Proofs are given for both halves (though the last sentence of the proof in the cohomology case requires a moment's thought to convince oneself of). The way to derive the cohomology spectr... | 11 | https://mathoverflow.net/users/23571 | 18590 | 12,387 |
https://mathoverflow.net/questions/18438 | 2 | Given a set of distances S, choose N unique points P on a number line such that the distances between the N points occur in S as much as possible. That is, maximize the occurence in S of the distances between the N points.
For example:
S = {2, 4}, N = 4
One answer would be P = {2, 4, 6, 8}, since the distances b... | https://mathoverflow.net/users/1214 | Given N points on a number line and m total distances between those points, are there efficent ways to optimize for particular values in m? | Some comments expanding on gowers' hunch: we may as well assume that the elements of S have gcd 1. If the size of the set is |S| = m and the sum of the elements of S is k then taking a block of N consecutive integers for $N \geq m$ gives (mN - k) pairs. There's also a theoretical maximum of $mN - \binom{m + 1}{2}$: eve... | 1 | https://mathoverflow.net/users/4658 | 18600 | 12,393 |
https://mathoverflow.net/questions/18588 | 18 | Then one can construct a model for the inverse limit by taking all the compatible sequences.
This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces are Hausdorff, then this is even a closed subspace.
However, if the spaces are not Hausdorff, it needn't be a closed... | https://mathoverflow.net/users/3969 | Is a inverse limit of compact spaces again compact ? | What does this example do ...
All spaces are on set $\{1,2,\dots\}$. Space $X\_n$ has topology that makes $\{1,2,\dots,n\}$ discrete and $\{n+1,\dots\}$ indiscrete. Of course $X\_n$ is compact non-Hausdorff. Map $X\_{n+1} \to X\_n$ by the "identity". Inverse limit is ... ???
| 34 | https://mathoverflow.net/users/454 | 18605 | 12,397 |
https://mathoverflow.net/questions/18608 | 7 | We would like to know how hard it is to count Eulerian orientation in an undirected 4-regular graph. For a given edge orientation to be Eulerian, we mean that every vertex has 2 in-edges and 2 out-edges.
It is known that counting Eulerian orientation in undirected graphs are #P-complete. We have tried to construct so... | https://mathoverflow.net/users/4547 | Counting Eulerian Orientation in a 4-regular undirected graph | Let $G$ be a planar graph. Consider a [medial graph](http://en.wikipedia.org/wiki/Knots_and_graphs#Medial_graph) $H=H(G)$, which is always $4$-regular. Often, problems about $G$ can be translated into the language of $H$ and vice versa. Closer to your question, the number of Eulerian orientations of $H$ is "almost" an ... | 10 | https://mathoverflow.net/users/4040 | 18621 | 12,410 |
https://mathoverflow.net/questions/18568 | 12 | Hello,
I am investigating the Leech lattice. Lately I have discovered following. Some lattices decompose into distinct set of orthonormal frames. For example E8 lattice which contains 240 unitary vectors in dimension 8 decompose into 15 sets of 16 vectors in each set. Each set contain 16 vectors of +- orthonormal bas... | https://mathoverflow.net/users/4714 | Leech lattice decomposition | Yes, such a decomposition exists. Here's a construction I learned from Elkies some time ago (it's mentioned in one of his papers, probably *Mordell-Weil lattices in characteristic 2, II*), using an action of the Gaussian integers Z[i] on the Leech lattice:
Let L be the Leech lattice, and consider the quotient L/(1+i)... | 18 | https://mathoverflow.net/users/4720 | 18626 | 12,415 |
https://mathoverflow.net/questions/18558 | 8 | I thought that I read a paper making this claim a few months ago, but now I can't find it. If the answer is yes, is there a nice way to go from the presentation of the right-angled coxeter group to a presentation of its right-angled artin subgroup? Thanks.
| https://mathoverflow.net/users/4027 | does every right-angled coxeter group have a right-angled artin group as a subgroup of finite index? | As James points out, the paper of Davis and Januskiewicz proves the inverse. To see that the answer to your question is 'no', consider the right-angled Coxeter group whose nerve graph is a pentagon. That is, it's the group with presentation
$\langle a\_1,\ldots, a\_5 \mid a\_i^2=1, [a\_i,a\_{i+1}]=1\rangle$
where the i... | 13 | https://mathoverflow.net/users/1463 | 18630 | 12,419 |
https://mathoverflow.net/questions/18636 | 18 | This question is inspired from [here](https://mathoverflow.net/questions/18547/number-of-unique-determinants-for-an-nxn-0-1-matrix), where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices have non-zero determinant?... | https://mathoverflow.net/users/2233 | Number of invertible {0,1} real matrices? | See [Sloane, A046747](http://oeis.org/A046747) for the number of singular (0,1)-matrices. It doesn't seem like there's an exact formula, but it's conjectured that the probability that a random (0,1)-matrix is singular is asymptotic to $n^2/2^n$.
Over $F\_2$ the probability that a random matrix is nonsingular, as $n \... | 22 | https://mathoverflow.net/users/143 | 18639 | 12,425 |
https://mathoverflow.net/questions/15282 | 18 | This is probably a very elementary question in symplectic geometry, a subject I've picked up by osmosis rather than ever really learning.
Suppose I have a symplectic manifold $M$. I believe that a *Lagrangian fibration* of $M$ is a collection of immersed Lagrangian submanifolds so that as a fibered manifold locally $... | https://mathoverflow.net/users/78 | To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N? | Your Question 1 is called Darboux theorem for fibrations (see: Arnold, V., Givental, A., Symplectic geometry, Dynamical Systems IV, Symplectic Geometry and its Applications (Arnold, V., Novikov, S., eds.), Encyclopaedia of Math. Sciences 4, Springer-Verlag, Berlin-New York, 1990.)
Here is how to construct suitable Da... | 14 | https://mathoverflow.net/users/2823 | 18645 | 12,430 |
https://mathoverflow.net/questions/18644 | 11 | This question is closely related to [this](https://mathoverflow.net/questions/13813/construction-of-the-stiefel-whitney-and-chern-classes) previous question.
Chern and Stiefel-Whitney classes can be defined on bundles over arbitrary base spaces. (In Hatcher's Vector Bundles notes, he uses the Leray-Hirsch Theorem, w... | https://mathoverflow.net/users/4042 | Uniqueness of Chern/Stiefel-Whitney Classes | I'm going to assume that your characteristic classes are supposed to live in the *singular cohomology* of the base space. Then to show your uniqueness result, it should be enough if you can produce, for any space $B$, a map $f:B'\to B$ such that $B'$ is paracompact, and $f$ induces an isomorphism in singular cohomology... | 17 | https://mathoverflow.net/users/437 | 18648 | 12,432 |
https://mathoverflow.net/questions/18653 | 7 | There is a beautiful way to see that the congruence subgroup $\Gamma(2)$ is free on two generators: the action of $\Gamma(2)$ on $\mathbb{H}$ is free and properly discontinuous, and there is a modular function $\lambda$ with respect to $\Gamma(2)$ coming from Legendre normal form such that $\mathbb{H}/\Gamma(2) \xright... | https://mathoverflow.net/users/290 | How does this geometric description of the structure of PSL(2, Z) actually work? | The key word here is "Bass-Serre Theory" -- using the action on the hyperbolic plane, you can easily cook up a nice action of $PSL\_2(\mathbb{Z})$ on a tree. This is all described nicely in Serre's book "Trees".
EDIT: Let me give a few more details. It turns out that a group $G$ splits as a free produce of two subgro... | 10 | https://mathoverflow.net/users/317 | 18655 | 12,437 |
https://mathoverflow.net/questions/18609 | 19 | One of the reasons why the classical theory of binary quadratic forms
is hardly known anymore is that it is roughly equivalent to the theory
of ideals in quadratic orders. There is a well known correspondence
which sends the $SL\_2({\mathbb Z})$-equivalence class of a form
$$ (A,B,C) = Ax^2 + Bxy + Cy^2 $$
with discr... | https://mathoverflow.net/users/3503 | Binary Quadratic Forms in Characteristic 2 | Starting, generally, with a commutative ring $R$ and a rank $2$ projective
module $P$ given with a quadratic form $\varphi$ we can form its Clifford algebra
$C(P)$. Its even part $S:=C^+(P)$ is then a commutative $R$ algebra of rank $2$.
If (which we may assume locally) $P=Re\_1+Re\_2$ we have that $S$ has basis
$1,e\_... | 13 | https://mathoverflow.net/users/4008 | 18657 | 12,438 |
https://mathoverflow.net/questions/18597 | 6 | Let $s\_{\lambda}$ and $m\_{\lambda}$ be the Schur and monomial symmetric functions indexed by an integer partition $\lambda$ ($\ell(\lambda)$ is the number of parts of $\lambda$ and $m\_i(\lambda)$ is the multiplicity of part $i$). By the hook-content formula we have:
$$
s\_{\lambda}(1^n) = \prod\_{u\in \lambda} \frac... | https://mathoverflow.net/users/771 | Is there a known formula for the number of SSYT of given shape with partition type? | Let $k(\lambda)=\sum\_\mu K\_{\lambda\mu}$. Then we have the generating function
$\sum\_\lambda k(\lambda)s\_\lambda = \prod\_{n\geq 1} (1-h\_n)^{-1}$.
| 9 | https://mathoverflow.net/users/2807 | 18658 | 12,439 |
https://mathoverflow.net/questions/18666 | 10 | Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
| https://mathoverflow.net/users/4619 | Continuous function from $[0,1]$ to $[0,1]$ | Yes. In fact, there exists such an $f$ taking every value *uncountably* many times.
Take a continuous surjection $g: [0, 1] \to [0, 1]^2$. (Such things exist: they're space-filling curves.) Then the composite $f$ of $g$ with first projection $[0, 1]^2 \to [0, 1]$ has the required property.
| 25 | https://mathoverflow.net/users/586 | 18668 | 12,441 |
https://mathoverflow.net/questions/17569 | 7 | The recent article found [here](http://arxiv.org/abs/1002.3622) revisits Thomason's proof that symmetric monoidal categories model all connective spectra, but stops short of showing that there is a full closed model structure on this category (as does, it seems, Thomason's original paper.) Is there such a thing?
My g... | https://mathoverflow.net/users/4466 | A Model Structure on Symmetric Monoidal Categories | One basic problem is that the category of symmetric monoidal categories isn't complete. Its completion, in a basic sense, is the category of multicategories, on which it seems reasonable to conjecture there is a model category structure whose homotopy category "is" the connective part of stable homotopy -- we hope to p... | 9 | https://mathoverflow.net/users/4732 | 18673 | 12,446 |
https://mathoverflow.net/questions/18661 | 5 | I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I have what I am hoping is a basic homotopy theory question:
A (orientable) PD-$n$ group is a group $G$ such that the Eil... | https://mathoverflow.net/users/4731 | PD3 groups and PD4 complexes | $G=Z^3$ is such an example. It is $\pi\_1(T^3)$ hence a PD-3 group. If $X$ is a Poincare 4-complex with fund group $Z^3$, then the injective (by Hopf) map on cohomology $H^2(G)\to H^2(X)$ cannot be onto, because its image is lagrangian for the intersection form by naturality of cup products. Dually $H\_2(X)\to H\_2(G)$... | 5 | https://mathoverflow.net/users/3874 | 18680 | 12,452 |
https://mathoverflow.net/questions/18667 | 3 | We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in the usual way. Here, I am talking about homology with coefficients in $\mathbb{Z}$. The group that we get is independent... | https://mathoverflow.net/users/2233 | Homology with Coefficients | Hi Tony.
This is not really a homology-question, the core of it is the fundamental group. The homomorphism you are using is used in the study of [Van Kampen diagrams](http://en.wikipedia.org/wiki/Van_Kampen_diagram). Consider a presentation $G=\langle A|R\rangle$. A Van Kampen diagram on $S$ is a labeled graph like y... | 4 | https://mathoverflow.net/users/3041 | 18684 | 12,455 |
https://mathoverflow.net/questions/18631 | 5 | What is the mirror manifold of the total space of the bundle $O(-1)\oplus O(-1)$ over $P^1$? I have tried to find the answer on the web but failed. Is there a good reference for this? Thanks.
| https://mathoverflow.net/users/2380 | Mirror of local Calabi-Yau | The physicists (see e.g. [this paper of Aganagic and Vafa](http://arxiv.org/abs/hep-th/0012041)) will write the mirror as a threefold $X$ which is an affine conic bundle over the holomorphic symplectic surface $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$ with discriminant a Seiberg-Witten curve $\Sigma \subset \mat... | 11 | https://mathoverflow.net/users/439 | 18711 | 12,474 |
https://mathoverflow.net/questions/18698 | 14 | Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of self-maps of $\Sigma$:
1) Earle and Eells proved that the components of $\text{Diff}(\Sigma)$ are contractible.
2) Ha... | https://mathoverflow.net/users/317 | Homotopy type of set of self homotopy-equivalences of a surface | A couple comments. For the result about diffeomorphism groups there is a very nice alternative proof due to A. Gramain in the Annales Scient. E.N.S. v.6 (1973), pp. 53-66, that uses no analysis, just basic differential topology. Another approach, which I'm not sure is written down anywhere in detail, is to take the pro... | 15 | https://mathoverflow.net/users/23571 | 18715 | 12,478 |
https://mathoverflow.net/questions/18677 | 8 | Can anyone provide me with the reference for the following fact
(idea of the proof will be appreciated too):
Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what the assumptions are: reductive complex algebraic group or maybe complex Lie group G with some restrictions. The cases I... | https://mathoverflow.net/users/2260 | Cohomology rings of $ GL_n(C)$, $SL_n(C)$ | If $G$ is a connected Lie group (or just a connected loop space with finite homology) then $H^\*(G,\mathbf{Q})$ is a Hopf algebra. Graded connected Hopf algebras over $\mathbf{Q}$ are always tensor products of exterior algebras in odd degrees with polynomial algebras in even degrees. Since polynomial algebras are infin... | 11 | https://mathoverflow.net/users/4183 | 18721 | 12,481 |
https://mathoverflow.net/questions/17980 | 0 | From what I understand, an exotic n-sphere is a manifold which is homeomorphic to the n-sphere but not diffeomorphic to it.
Now I have read that there are no exotic 2-spheres.
But isn't something like a tetrahedron an example of a manifold which is homeomorphic to the sphere, but not diffeomorphic ? (Because of the cor... | https://mathoverflow.net/users/4279 | Question about exotic spheres | A exotic sphere is (by definition) a differentiable manifold. So if you want to consider the tetrahedron, you have to specify, what differentiability at one of the edges means. As soon as you specified this, you will just get the 2-sphere.
| 5 | https://mathoverflow.net/users/3969 | 18724 | 12,482 |
https://mathoverflow.net/questions/18723 | 14 | Often, certain symbols in mathematics denote different things in different fields. Is there any sort of ordered list that will tell you what a certain symbol means in alphabetical order by the symbol's alias in LaTeX, perhaps with the way to pronounce it out loud?
I'm thinking of something like this [Wikipedia page]... | https://mathoverflow.net/users/1353 | Mathematical symbols, their pronunciations, and what they denote: Does a comprehensive ordered list exist? | Comments suggest this ...
<http://www.fileformat.info/info/unicode/category/Sm/list.htm>
However, a designation like "rightwards arrow above reverse tilde operator" doesn't really answer the question here, does it?
| 5 | https://mathoverflow.net/users/454 | 18731 | 12,488 |
https://mathoverflow.net/questions/18701 | 7 | Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T\_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t\_f$ such that there exists a [Fermi coordinate system](http://en.wikipedia.org/wiki/Fermi_coordinates) adapted to $\gamma$ up to time $t\_f$.
... | https://mathoverflow.net/users/238 | Existence of Fermi coordinates on a Riemannian manifold | I would answer this question this way:
First, without any further assumptions about the Riemannian metric on $M$, you don't even have a lower bound on the time $t\_f$ for which the geodesic *exists* and does not intersect itself. The geodesic may fail to exist either simply because the metric is incomplete.
Second,... | 6 | https://mathoverflow.net/users/613 | 18737 | 12,492 |
https://mathoverflow.net/questions/18734 | 3 | I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this [question](https://mathoverflow.net/questions/17612/finding-saturated-open-sets)).
A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets are compact. It is called quotient map, iff a sub... | https://mathoverflow.net/users/3969 | Is a proper quotient map closed ? | No. Let X={1,2,3} and Y={1,2}. Let f map 1 to 1, 2 and 3 to 2. Let the topology on X be {∅,{2},{1,2},{2,3},{1,2,3}} and that on Y be {∅,{2},{1,2}}. f maps the closed set {3} onto the non-closed set {2}.
| 3 | https://mathoverflow.net/users/802 | 18742 | 12,496 |
https://mathoverflow.net/questions/18719 | 3 | Let $ K $ be a finite extension of a $p$-adic field or a number field, L a finite extension of $K$. The following fact holds: $
\text{Gal}(K^{\text{ab}} / K) \rightarrow \text{Gal}(L^{\text{ab}} / L)
$, where the arrow is the Transfer (Verlagerung) map, is injective.
I wonder whether this is an arithmetic fact or a f... | https://mathoverflow.net/users/2701 | Injectivity of Transfer (Verlagerung) map | I'm not sure how to answer the more philosophical question (it's likely you could encode enough of the axioms to force the purely group-theoretic version of the question to be true, but to ask whether that's what's "really" going on....), but it's certainly not true for all pairs of groups that fit in a similar commuta... | 5 | https://mathoverflow.net/users/35575 | 18745 | 12,498 |
https://mathoverflow.net/questions/18748 | 8 | Let's say we want to define a choice function for certain particular subsets $S \subset2^{\mathbb{R}}$, i.e. we want a function $c:S \rightarrow \mathbb{R}$ such that $c(X)\in X$ for every $X\in S$. We don't want to invoke the axiom of choice. Clearly we require $\emptyset\notin S$.
For example, if $S$ is the set of ... | https://mathoverflow.net/users/1049 | Choice function for Borel sets? | It is impossible without using the Axiom of Choice to prove the existence of such a choice function for $F\_{\sigma}$
sets, which include the countable sets. To see this, it is easier to work with the space $2^{\mathbb{N}}$ of infinite binary sequences instead of $\mathbb{R}$. Let $E$ be the eventual equality equivalen... | 6 | https://mathoverflow.net/users/4706 | 18755 | 12,501 |
https://mathoverflow.net/questions/18747 | 15 | Well, I don't have any notion of *arithmetic geometry*, but I would like to understand what arithmetic geometers mean when they say "integer point of a variety/scheme $X$" (like e.g. in "integer points of an elliptic curve").
Is an integer point just defined as a morphism from $Spec\mathbb{Z}$ into $X$?
Suppose $X$... | https://mathoverflow.net/users/4721 | Integer points (very naive question) | To make sense of the notion of integer points, your scheme should be defined over $\mathbb{Z}$. What do we mean by that? Of course we should not ask for a structure map tp $Spec(\mathbb{Z})$, since every scheme has one such map. The right notion is the following.
Let $X$ be a scheme over $\mathbb{C}$; so by definitio... | 22 | https://mathoverflow.net/users/828 | 18757 | 12,502 |
https://mathoverflow.net/questions/18756 | 0 | Consider the [2D Shubert function](http://www.staff.brad.ac.uk/jpli/research/scga/shubert/shubrt21.htm). As given in that page, the function has 18 global minima and several local minima. How can I find the (x,y) of all these minima? Any help appreciated. If it was a summation (instead of a product), I would have done ... | https://mathoverflow.net/users/3552 | Finding the local/global minima of Shubert function | At Jacques' cajoling, I'm turning the comments into an answer.
The two dimensional Shubert function is just the product of the one dimensional one by itself. $f(x,y) = g(x)g(y)$ where $g(x) = \sum\_{j = 1}^5 j \cos( (j+1)x + j)$ is the 1 dimensional Shubert function. Observe that the local maxima are all positive, a... | 2 | https://mathoverflow.net/users/3948 | 18762 | 12,505 |
https://mathoverflow.net/questions/18772 | 12 | Suppose $G$ is a semisimple group, and $V\_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the multiplicity of the trivial representation in $V\_{\lambda}|\_{H}$? Is there a simple way to read this off from $\lambda$ and... | https://mathoverflow.net/users/1464 | Occurrence of the trivial representation in restrictions of Lie group representations | The short answer is: no. In theory, there are explicit formulae for branchings based on the Weyl character formula, but no reasonable person would call these simple.
There are interesting results which give you some information about branching multiplicities, but all involve real work. For instance, if G and H are co... | 11 | https://mathoverflow.net/users/66 | 18779 | 12,513 |
https://mathoverflow.net/questions/18774 | 6 | Is there some **onto** function $f:$ $\mathbb{C}$ $\rightarrow$ $\mathbb{C}$
such that for each triangle $T$ (with its interior), $f(T)$ is a
square (with interior, too) ?
I would have the same question for triangles and squares *without* interior, respectively.
| https://mathoverflow.net/users/4750 | Triangles, squares, and discontinuous complex functions | With interior: yes. Fix a sequence of squares $Q\_1\subset Q\_2\subset\dots$ whose union is the entire plane. Then arrange a map $g:\mathbb R\to\mathbb R^2$ such that, for every nontrivial segment $[a,b]\subset\mathbb R$, its image is one of the squares $Q\_i$. To do that, construct countably many disjoint Cantor sets ... | 10 | https://mathoverflow.net/users/4354 | 18783 | 12,515 |
https://mathoverflow.net/questions/18716 | 20 | I had been looking lately at Sylow subgroups of some specific groups and it got me to wondering about *why* Sylow subgroups exist. I'm very familiar with the proof of the theorems (something that everyone learns at the beginning of their abstract algebra course) -- incidentally my favorite proof is the one by Wielandt ... | https://mathoverflow.net/users/2784 | Sylow Subgroups | Victor, you should check out Sylow's paper. It's in Math. Annalen 5 (1872), 584--594. I am looking at it as I write this. He states Cauchy's theorem in the first sentence and then says "This important theorem is contained in another more general theorem: if the order is divisible by a prime power then the group contain... | 20 | https://mathoverflow.net/users/3272 | 18788 | 12,517 |
https://mathoverflow.net/questions/18787 | 21 | Here's a question I can't answer by myself: The Reflection Principle in Set Theory states for each formula $\phi(v\_{1},...,v\_{n})$ and for each set M there exists a set N which extends M such that the following holds
$\phi^{N} (x\_{1},...,x\_{n})$ iff $\phi (x\_{1},...,x\_{n})$ for all $x\_{1},...x\_{n} \in N$
Th... | https://mathoverflow.net/users/4753 | Montague's Reflection Principle and Compactness Theorem | For any finite set of axioms K of ZFC, ZFC proves "K has a model", via the reflection principle as you note. However, ZFC does not prove "for any finite set of axioms K of ZFC, K has a model". The distinction between these two is what prevents ZFC from proving that ZFC has a model.
(That is, even though, as you note,... | 28 | https://mathoverflow.net/users/3902 | 18793 | 12,520 |
https://mathoverflow.net/questions/18753 | 14 | Consider a compact manifold *M*. For a vector field *X* on *M*, let $\phi\_X$ denote the diffeomorphism of *M* given by the time 1 flow of *X*.
If *X* and *Y* are two vector fields, is $\phi\_X \circ \phi\_Y$ necessarily of the form $\phi\_Z$ for some vector field *Z*?
Since $X\mapsto \phi\_X$ can be thought of as ... | https://mathoverflow.net/users/4747 | Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold? | To answer your first question, the composition of two time-1 flows won't necessarily be another time-1 flow.
One way to see this is to note that when a time-1 flow $\phi\_X$ has a periodic point $P$ (period > 1), then $P$ can't be hyperbolic since it lies on a closed orbit of the flow for $X$. (The eigenvector of $D\... | 12 | https://mathoverflow.net/users/1227 | 18801 | 12,524 |
https://mathoverflow.net/questions/18799 | 39 | The push-pull formula appears in several different incarnations. There are, at least, the following:
1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ we have $f\_{\*} (\mathcal{F} \otimes f^{\*} \mathcal{G}) \cong f\_{\*} (\mathcal{F}) \otimes \mathcal{G}$.
... | https://mathoverflow.net/users/828 | Ubiquity of the push-pull formula | [This paper](http://www.math.uiuc.edu/K-theory/0573/) by Fausk, Hu and May does not exactly tell you why those maps should be isomorphisms in more concrete situations, but it cleanly explains the abstract settings in which they arise - look e.g. at Propositions 2.4 and 2.8 for equivalent formulations of projection form... | 7 | https://mathoverflow.net/users/733 | 18814 | 12,535 |
https://mathoverflow.net/questions/18813 | 19 | This question may end up [closed], but I'm going to ask and let the people decide. It's certainly the kind of question that I'd ask people at tea, and it's not one whose answer I've been able find with Google.
TeX, I have heard, is Turing complete. In theory, this means that we can do modular arithmetic with LaTeX pr... | https://mathoverflow.net/users/35508 | Modular Arithmetic in LaTeX | Get a current version of Ti*k*Z and use `\pgfmathmod`!
| 25 | https://mathoverflow.net/users/1409 | 18815 | 12,536 |
https://mathoverflow.net/questions/18817 | 9 | Is it true that for every integer $r$, the equation $2^m = 3^n + r$ has at most a finite number of integer solutions? I understand that this is a special case of Pillai's conjecture, which is unsolved.
If the statement is true, then can we verify the finiteness of the solution set using modular arithmetic? To be prec... | https://mathoverflow.net/users/4758 | Does 2^m = 3^n + r have finitely many solutions for every r? | Yes, it is true that this kind of equation ax+by=c, where a,b,c are non-zero and fixed and x,y are allowed to only have prime factors in a finite set, has only finitely many solutions. This is a special case of Siegel's theorem on integral points on curves.
Your second question may be unknown in the generality you p... | 15 | https://mathoverflow.net/users/2290 | 18819 | 12,539 |
https://mathoverflow.net/questions/18794 | 44 | I recently got interested in game theory but I don't know where should I start.
Can anyone recommend any references and textbooks?
And what are the prerequisites of game theory?
| https://mathoverflow.net/users/3124 | How to start game theory? | "[A course in game theory](http://theory.economics.utoronto.ca/books/)" by Martin J. Osborne and Ariel Rubinstein is probably the standard more mathematical starting point. A more concise, more modern, and slightly CS-leaning text is "[Essentials of Game Theory -- A Concise, Multidisciplinary Introduction](http://www.g... | 14 | https://mathoverflow.net/users/4762 | 18829 | 12,548 |
https://mathoverflow.net/questions/18764 | 7 | I am looking for a reference for the following well-known fact: For any subdiagram $\Delta\_0$ of of the Dynkin diagram $\Delta=D(G)$
of an adjoint simple group $G$ over an algebraically closed field $k$, there exists a reductive subgroup of maximal rank $G\_0\subset G$
with Dynkin diagram $\Delta\_0$.
To be more pre... | https://mathoverflow.net/users/4149 | Reductive subgroup corresponding to a subdiagram of the Dynkin diagram of a simple group | Dear Mikhail,
If I understand correctly, your Lemma 2 is implied by
SGA 3 Exposé 22, Théorème 5.4.7.
Everything is on a general base S (that you may take as your algebraically closed field). The kind of subgroup you want is called "de type (R)" (see Définition 5.2.1) and a subset of R that corresponds to such a gro... | 10 | https://mathoverflow.net/users/4763 | 18830 | 12,549 |
https://mathoverflow.net/questions/18594 | 3 | Consider the following
$f\_{t+1}(z)=p\_{12} f\_{t}(z/A)+ p\_{21} f\_{t}(z/B)+p\_{22} f\_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f\_t$ is a probability distribution. Are there any nice distribution families that are preserved under the transformation? Fail that, are there $f\_t$ such that $f\_{t... | https://mathoverflow.net/users/4126 | Finding a distribution family that is preserved under mixture. | One can rewrite the problem in terms of products of i.i.d. random variables as follows.
Assume that $X\_t$ has distribution density $f\_t$. Then, the relation between $f\_t$ and $f\_{t+1}$ means that one can choose $X\_{t+1}=X\_tZ\_{t+1}$, where the $Z\_t$ are i.i.d. and $Z\_t=A$ or $B$ or $A+B$, with probabilities ... | 2 | https://mathoverflow.net/users/4661 | 18835 | 12,552 |
https://mathoverflow.net/questions/18844 | 6 | To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric is complete, and if a noncompact manifold has finite topological type(ie is diffeomorphic to the interior of a compact ... | https://mathoverflow.net/users/2497 | Does every smooth manifold of infinite topological type admit a complete Riemannian metric? | By Whitney embedding theorem any smooth manifold embeds into some Euclidean space as a closed subset. The induced metric is complete.
In fact, a good exercise is to show that any Riemannian metric is conformal to a complete metric.
| 18 | https://mathoverflow.net/users/1573 | 18851 | 12,561 |
https://mathoverflow.net/questions/18798 | 2 | This question is related to [this previous question](https://mathoverflow.net/questions/18758/drawing-a-combinatorial-3-configuration-of-points-and-lines-with-pseudolines). Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in ... | https://mathoverflow.net/users/607 | Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines? | The fist part of your question has a negative answer, since both Fano plane (73) and Moebius-Kantor configuration (83) are *cyclic configurations*. [Here](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-4520GM0-F&_user=10&_coverDate=02%252F06%252F2002&_alid=1258918945&_rdoc=32&_fmt=high&_orig=search&_cdi... | 1 | https://mathoverflow.net/users/4400 | 18858 | 12,567 |
https://mathoverflow.net/questions/18846 | 1 | I can't understand this sentence i the article of Majid "Tannaka-Krein theorem for quasi-Hopf algebras and other results" about the reconstruction of a quasi-algebra (in fact its dual) from a given braided category (C,c) with a forgetful functor F ; at the end it considers the action of Drinfeld-twisting upon quasi-alg... | https://mathoverflow.net/users/4770 | Why does twisting quasi-Hopf algebras work (as in majid's article) | You might find these notes helpful as a second source to read fro:
<http://www-math.mit.edu/~etingof/tenscat1.pdf>
There is a careful elaboration of the various variants of Tannaka-Krein construction, and the case of quasi-bialgebras and quasi-Hopf algebras is discussed. So far as I recall, Drinfeld twisting of a q... | 3 | https://mathoverflow.net/users/1040 | 18861 | 12,568 |
https://mathoverflow.net/questions/18854 | 3 | The classification theorem for surfaces says that the complete set of homeomorphism classes of surfaces is
{ $S\_g : g \geq 0$ } $ \cup$ { $N\_k : k \geq 1$ },
where $S\_g$ is a sphere with $g$ handles, and $N\_k$ is a sphere with $k$ crosscaps. The first homology groups are easy to compute. They are $H\_1 (S\_g) ... | https://mathoverflow.net/users/2233 | Homology of Surfaces with Holes | Autumn's answer captures the essence why there is a $\mathbb{Z}\_2$ is in the first homology
of a closed nonorientable surface.
If you remove a disk from a closed surface, the resulting object has a $1$-dimensional $CW$-complex as a strong deformation retract, so that the homology of the resulting object
has no torsi... | 6 | https://mathoverflow.net/users/4304 | 18862 | 12,569 |
https://mathoverflow.net/questions/10730 | 8 | There is a textbook theorem that the categories of unipotent algebraic groups and nilpotent finite-dimensional Lie algebras are equivalent in characteristic zero. Indeed, the exponential map is an algebraic isomorphism in this case and the group structure can be defined in terms of the Lie algebra structure and vice ve... | https://mathoverflow.net/users/2106 | References for theorem about unipotent algebraic groups in char=0? | Demazure-Gabriel, *Groupes algebriques*, Tome I (published in 1970) is a more explicit source, if available. Chapitre IV treats "groupes affines,
nilpotents, resolubles", while Chapitre V specializes to commutative affine groups.
Typically they work over an (almost) arbitrary field $k$, but IV.2.4 is
devoted to "groupe... | 7 | https://mathoverflow.net/users/4231 | 18869 | 12,575 |
https://mathoverflow.net/questions/18840 | 23 | I wish to learn nonstandard analysis. Are there any good book recommendations? I'm familiar with the ZFC system, and learnt analysis the classical way. I've found some undergraduate texts, but they are too verbose.
If there are applications to complex or functional analysis, that would be great.
Thanks in advance.... | https://mathoverflow.net/users/nan | nonstandard analysis book recommendation | This one sounds like what you want:
[Arkeryd, Cutland and Henson: Nonstandard Analysis, Theory and Applications.](http://books.google.co.uk/books?id=X0BbnEP3jN0C&pg=PA1&lpg=PA1&dq=nonstandard+analysis+cutland&source=bl&ots=B_xdfk9wsQ&sig=iIZA61f8iIvbpE1Wdap0azWtC-c&hl=en&ei=rx6lS_3hEo2OjAeX8tnzCQ&sa=X&oi=book_result&... | 10 | https://mathoverflow.net/users/733 | 18872 | 12,578 |
https://mathoverflow.net/questions/9915 | 14 | If G is a reductive algebraic group (say over ℂ), T a maximal torus, then we can consider its Weyl group W which acts on the abelian group Y of one parameter subgroups of T. Thus we may form the semidirect product, which I will call the affine Weyl group.
In the semisimple simply connected case, this affine Weyl grou... | https://mathoverflow.net/users/425 | Affine Weyl groups as Coxeter groups | In the abstract Bourbaki set-up, the affine Weyl group is defined to be a
semidirect product of an *irreducible* Weyl group with its coroot lattice.
This is naturally a Coxeter group, characterized in terms of its
positive semidefinite Coxeter matrix. The basic theory is developed
independently of applications in Lie t... | 14 | https://mathoverflow.net/users/4231 | 18875 | 12,580 |
https://mathoverflow.net/questions/18876 | 2 | There is the invariant Maurer–Cartan 1-form on a compact semi-simple Lie group G. So if we pull it back using a map from X to G then we get a G-connection on X. The question is, can all G-connections on X ( here just regarded as elements in ${\Omega}^1(X, \mathfrak{g})$, i.e. we are using the trivial G-bundle over X ) ... | https://mathoverflow.net/users/4782 | Can all G-connections on a Riemann surface X be induced by maps from X to G | José Figueroa-O'Farrill has already pointed out one necessary condition, namely that your connection must be flat. The remaining condition is that the monodromy should be trivial. In what follows $X$ is any connected smooth manifold, not necessarily a surface, and $G$ is any Lie group.
Let's first consider the analog... | 7 | https://mathoverflow.net/users/250 | 18885 | 12,586 |
https://mathoverflow.net/questions/18893 | 6 | The idea hit me when I was in my Elliptic Curve Cryptography class. $Z\_n \leftrightarrow Z\_{f\_1} \times Z\_{f\_2} \times ...$ where $f\_1 \times f\_2 \times ... = n$ and $\{f\_1, f\_2, ...\}$ are pairwise coprime. Applications of this Chinese Remainder Theorem not only include computational speedups (in the case of ... | https://mathoverflow.net/users/3737 | Analog to the Chinese Remainder Theorem in groups other than Z_n. | The Chinese Remainder theorem is usually thought of as an isomorphism of *rings*, not just of cyclic groups. In this regard it has a vast generalization:
Theorem (Ideal-theoretic CRT): Let R be a commutative ring, and let $I\_1,\ldots,I\_n$ be a finite set of ideals in $R$ which are **pairwise comaximal**: for all $i... | 20 | https://mathoverflow.net/users/1149 | 18899 | 12,596 |
https://mathoverflow.net/questions/18877 | 18 | 1. Does ${\mathbb R}$ have proper, countable index subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care if it contains $1$.) By index, I mean index as an additive subgroup.
2. Given some real number $x$, when is it possibl... | https://mathoverflow.net/users/4783 | Are there countable index subrings of the reals? | Perhaps surprisingly, it turns out that such subrings do exist. This was proved in Section 2 of my paper:
Simon Thomas, Infinite products of finite simple groups II,
J. Group Theory 2 (1999), 401--434.
The basic idea of the proof is quite simple. Clearly the ring of $p$-adic integers has countable index in the fie... | 28 | https://mathoverflow.net/users/4706 | 18900 | 12,597 |
https://mathoverflow.net/questions/18903 | 7 | In the article ['André Weil As I Knew Him'](http://www.ams.org/notices/199904/shimura.pdf) in the April 1999 issue of Notices of the AMS, Shimura recounts how in 1996 André Weil (then 90 years old) didn't remember a mistake of Minkowski. Specifically, quoting from mid-paragraph:
*"... In fact, to check that point, I ... | https://mathoverflow.net/users/2604 | What is Shimura referring to by "an incorrect formula given by Minkowski... known to most experts." | I believe he is referring to the result in Minkowski's dissertation that gives a formula for the mass (in the number/lattice theory sense) over a genus, that is, the sum of reciprocals of the group orders of all inequivalent quadratic forms in a genus. ([wikipedia](http://en.wikipedia.org/wiki/Smith%E2%80%93Minkowski%E... | 12 | https://mathoverflow.net/users/239 | 18907 | 12,602 |
https://mathoverflow.net/questions/18916 | 6 | Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}\_q(x)$ (function fields in 1 variable over a finite field). The latter are isomorphic to the category of curves over $\mathbb{F}\_q(x)$, and they can be generalized to function fields in $n$ variabl... | https://mathoverflow.net/users/4692 | Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field? | The function fields (in one or more variables) over $\mathbb{F}\_q$ are precisely the infinite, finitely generated fields of characteristic $p$. Thus an at least reasonable characteristic $0$ analogue is given by the (necessarily infinite!) finitely generated fields of characteristic $0$. In other words, function field... | 5 | https://mathoverflow.net/users/1149 | 18917 | 12,609 |
https://mathoverflow.net/questions/18926 | 6 | Let $X\_\bullet \longrightarrow Y\_\bullet \longleftarrow Z\_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like).
On p. 14-9 of [these notes](http://www.math.washington.edu/~warner/TTHT_Warner.pdf) there is an example which shows that if $Y\_\bullet$ and $Z\_\bullet$ are levelwise connected t... | https://mathoverflow.net/users/318 | Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander | The following represents what I know about this; I don't know of a published reference.
Given a map $f:Z\_\bullet\to Y\_\bullet$ of simplicial "spaces" (to make things easy, assume spaces are simplicial sets), let's call it a realization quasi-fibration (RQF) if for every $U\_\bullet\to Y\_\bullet$, the homotopy pull... | 7 | https://mathoverflow.net/users/437 | 18927 | 12,614 |
https://mathoverflow.net/questions/18935 | 1 | I'm trying to teach myself category theory from Steve Awodey's [Category Theory](http://www.andrew.cmu.edu/course/80-413-713/notes/). Chapter 2 asserts:
>
> It is not hard to see that a filter F is an ultrafilter just if for every element b ∈ B, either b ∈ F or ¬b ∈ F, and not both (exercise!).
>
>
>
I've mana... | https://mathoverflow.net/users/4793 | F is ultrafilter over a Boolean algebra implies that for every b, either b or not-b is in F? | Assume you have a proper filter $F$ that avoids both $b$ and $\neg b$. Then, you could consider the filter generated by $F\cup\{b\}$ - which is to say the smallest filter $F'$ containing $F$ and $b$.
Since $F$ was a proper filter it follows that $0\not\in F$.
If $0\in F'$, then this means that there is some $f\in... | 4 | https://mathoverflow.net/users/102 | 18940 | 12,622 |
https://mathoverflow.net/questions/18928 | 11 | Elementary commutative algebra fact: for two proper ideals I and J of a commutative ring R, we have $V(IJ)=V(I\cap J)=V(I)\cup V(J)$.
Closed subschemes are related to sheaves of ideals. There is operation of intersection and product between sheaves of ideals, which is similar to the affine case.
I see in many place... | https://mathoverflow.net/users/2666 | Union of closed subschemes with the structure sheaf over it | Because the first one is the right answer in the case of affine varieties, and the second one is not. Indeed, $R/I$, $R/J$ are nilpotent-free implies $R/I\cap J$ is nilpotent-free, but not so for $R/IJ$.
| 10 | https://mathoverflow.net/users/1784 | 18942 | 12,624 |
https://mathoverflow.net/questions/18939 | 1 | Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x\_1,...x\_n]$ (resp. $k[y\_1,...,y\_m]$).
Let $Z$ be the affine scheme defined by the ideal $L$ in $k[z\_1,...z\_s]$, and let $f^\\*:k[z]/L\rightarrow k[x]/I$ (resp. $g^\\*:k[z]/L\rightarrow k[y]/J$) be $k$-homo... | https://mathoverflow.net/users/4721 | Expressing fiber product of affines via an ideal | Dear unknown, let me first congratulate you on the clearness of your question and the quality of your notation, which I'm now going to use.
The fibre product $X\times\_Z Y$ is the subscheme of $\mathbb A\_k^n \times \mathbb A\_k^m$ described by an ideal $\mathfrak A \subset k[x,y]$.
That ideal is $\mathfrak A=I^e + J... | 6 | https://mathoverflow.net/users/450 | 18947 | 12,626 |
https://mathoverflow.net/questions/18922 | 6 | Are there books or survey articles explaining the subject to a non-expert? To clarify what I mean, here is a couple of issues that I would like to read about. (I am mainly interested in references but would appreciate answers to these specific questions.)
1) As far as I remember, PA do not have a "built-in" scheme fo... | https://mathoverflow.net/users/4354 | A book explaining power and limitations of Peano Axioms? | For obvious reasons, foundations textbooks (undergraduate or beginning graduate level) tend to have essentially no prerequisites, so I suspect you will find most of them to be accessible.
Here are some higher level recommendations that are more to the point. These assume some rudimentary knowledge of Computability Th... | 8 | https://mathoverflow.net/users/2000 | 18950 | 12,628 |
https://mathoverflow.net/questions/18847 | 39 | A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally then left for the bulk of the course, in terms of defining singular homology, proof of the harder Eilenberg-Steenrod axio... | https://mathoverflow.net/users/360 | "Homotopy-first" courses in algebraic topology | I was a heavily involved TA for such a graduate course in 2006 at UC Berkeley.
We started with a little bit of point-set topology introducing the category of compactly generated spaces. Then we moved into homotopy theory proper. We covered CW-complexes and all the fundamental groups, Van-Kampen's Theorem, etc. From ... | 30 | https://mathoverflow.net/users/184 | 18955 | 12,631 |
https://mathoverflow.net/questions/18957 | 7 | SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically permutes the last three rows of the matrix A. I wont define h here but, except for the signs of the terms, it is the us... | https://mathoverflow.net/users/4796 | Sarrus determinant rule: references, extensions | I have found references to the Sarrus determinant rule and to an extension of it. In The Quarterly journal of pure and applied mathematics, Volume 38 which is available on Google books in the article "A fourth list of writings on determinants". On page 239 There is a reference to what I believe is Sarrus original resul... | 1 | https://mathoverflow.net/users/1098 | 18961 | 12,634 |
https://mathoverflow.net/questions/18959 | 12 | This is a followup to [Analog to the Chinese Remainder Theorem in groups other than Z\_n.](https://mathoverflow.net/questions/18893/analog-to-the-chinese-remainder-theorem-in-groups-other-than-z-n) . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese Remainder Th... | https://mathoverflow.net/users/2530 | Chinese Remainder Theorem for rings: why not for modules? | The second result you're talking about is also sometimes called the Chinese remainder theorem, and can be derived from the Chinese remainder theorem for rings by "tensoring the CRT isomorphism" with $A$. Explicitly, (1) gives
$R/\prod\_{k=1}^n I\_k\cong\prod\_{k=1}^n R/I\_k$
via the natural map. This is an isomorph... | 16 | https://mathoverflow.net/users/4351 | 18962 | 12,635 |
https://mathoverflow.net/questions/18964 | 17 | Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much and Grothendieck's methods are used instead. Is this because it is felt that Dwork's method is not general or powerful eno... | https://mathoverflow.net/users/4692 | Dwork's use of p-adic analysis in algebraic geometry | The premise of the question is not correct. Dwork's methods (and modern descendants of them) are a major part of modern arithmetic geometry over $p$-adic fields, and of $p$-adic analysis.
You could look at the two volume collection *Geometric aspects of Dwork theory* to get
some sense of these developments.
Just to ... | 33 | https://mathoverflow.net/users/2874 | 18966 | 12,637 |
https://mathoverflow.net/questions/18960 | 1 | We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions.
We want to calculate the winding number of this curve around the origin.
The most natural way to do it is to calculate the path integral:
$$\int\_C \frac{-y\,dx+x\,dy}{x^2+y^... | https://mathoverflow.net/users/4797 | How To Calculate A Winding Number? | This is simple if you can draw a picture of your curve. Find a direction so that your tangent is always moving as you pass through it. Count the number of tangents pointing in that direction with a sign. +1 if you are moving through the direction counterclockwise, and -1 if you are moving through the direction clockwis... | 7 | https://mathoverflow.net/users/4304 | 18968 | 12,639 |
https://mathoverflow.net/questions/11713 | 11 | The slice-ribbon conjecture asserts that all slice knots are ribbon.
This assumes the context:
1) A `knot' is a smooth embedding $S^1 \to S^3$. We're thinking of the 3-sphere as the boundary of the 4-ball $S^3 = \partial D^4$.
2) A knot being slice means that it's the boundary of a 2-disc smoothly embedded in ... | https://mathoverflow.net/users/1465 | slice-ribbon for links (surely it's wrong) | Ryan, I think this is an open problem. The best related result I know is a theorem of Casson and Gordon [A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983)] saying that for a fibred knot that bounds a homotopically ribbon disk in the 4-ball, the slice complement is also fibred.
More p... | 12 | https://mathoverflow.net/users/4625 | 18971 | 12,641 |
https://mathoverflow.net/questions/18974 | 2 | Let $G$ be a finitely presented group and $N$ a normal subgroup. Is $N$ finitely generated or normally finitely generated? Here normally finitely generation means that for some finite set $S$ of elements in $N$, every element of $N$ can be writen as a product of $G$-conjugation of elements in $S$. Thanks.
| https://mathoverflow.net/users/1546 | Is a normal subgroup of a finitely presented group finitely generated or normal finitely genrated? | If $G$ is the free group on two generators, then $N$ the commutator subgroup is not finitely generated.
If $H$ is any finitely generated, but not finitely presented group, then $H$ is the quotient of a finitely generated free group $G$, with kernel $N$ which is not normally finitely generated.
Steve
| 11 | https://mathoverflow.net/users/1446 | 18975 | 12,644 |
https://mathoverflow.net/questions/18790 | 10 | For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at zero of the Killing form applied to two elements (i.e. $t^k\mathfrak{g}$ and $t^{-k-1}\mathfrak{g}$ are paired by the Killi... | https://mathoverflow.net/users/66 | Is there a good reference for the relationship between the Yangian and formal based loop group? | I will suggest the article [Quantization of Lie bialgebras, III](http://arxiv.org/abs/q-alg/9610030/) by Pavel Etingof and David Kazhdan. It discusses both the Yangian and the dual Yangian as examples in the context of quantization of Lie bialgebras of functions on a curve with punctures.
Also, so far as I know, no-o... | 5 | https://mathoverflow.net/users/3316 | 18983 | 12,648 |
https://mathoverflow.net/questions/18987 | 27 | Using Alexander duality, you can show that the Klein bottle does not embed in $\mathbb{R}^3$. (See for example [Hatcher's book Chapter 3](http://www.math.cornell.edu/~hatcher/AT/ATch3.pdf) page 256.) Is there a more elementary proof, that say could be understood by an undergraduate who doesn't know homology yet?
| https://mathoverflow.net/users/3557 | Why can't the Klein bottle embed in $\mathbb{R}^3$? | If you are willing to assume that the embedded surface $S$ is polyhedral, you can prove that it is orientable by an elementary argument similar to the proof of polygonal Jordan Theorem. Of course the proof is translation of a homology/transversality/separation argument.
Fix a direction (nonzero vector) which is not p... | 24 | https://mathoverflow.net/users/4354 | 19005 | 12,663 |
https://mathoverflow.net/questions/18982 | 8 | In a letter to Tate from 1987, Serre describes a beautiful Theorem relating mod p modular forms to quaternions ("Two letters on quaternions and modular forms (mod p)", Israel J. Math. 95 (1996), 281--299). At the beginning of Remark (4) on page 284, Serre says that every supersingular elliptic curve E over \bar F\_p ha... | https://mathoverflow.net/users/4800 | Supersingular elliptic curves and their "functorial" structure over F_p^2 | (**EDIT:** I've rewritten my argument in terms of the inverse functor, i.e., base extension, since it is clearer and more natural this way.)
Much of what is below is simply a reorganization of what Robin Chapman wrote.
>
> **Theorem:** For each prime $p$, the base extension functor from the category $\mathcal{C}\... | 9 | https://mathoverflow.net/users/2757 | 19013 | 12,668 |
https://mathoverflow.net/questions/19041 | 11 | Is there any possibility of a Poisson Geometry version of the Fukaya category? Given a Poisson manifold Y, objects could be manifolds with isolated singularities X which have the property that TX is contained in NX maximally. The naive example would be something like the Poisson structure on R^2 which is (x^2 + y^2)(d/... | https://mathoverflow.net/users/6986 | A Poisson Geometry Version of the Fukaya Category | The fundamental technique of symplectic topology is the theory of pseudo-holomorphic curves. One studies maps $u$ from a Riemann surface into a symplectic manifold, equipped with an almost complex structure tamed by the symplectic form, such that $Du$ is complex-linear. Numerous algebraic structures can be built from s... | 6 | https://mathoverflow.net/users/2356 | 19044 | 12,683 |
https://mathoverflow.net/questions/18970 | 8 | When I read descriptions of Apollonius' treatise on conics, some of them say that he invented co-ordinate geometry, some say that he kind of did and others are silent on the matter. Or is it the case that there is no simple answer?
| https://mathoverflow.net/users/4692 | Did Apollonius invent co-ordinate geometry? | Let V be the vertex of a parabola, F its focus, X a point on its symmetry axis, and A a point on the parabola such that AX is orthogonal to VX. It was well within the power of the Greeks to prove relations such as $VX:XA = XA:4VF$. If you introduce coordinate axes, set $x = VX$, $y = XA$ and $p = VF$, you get $y^2 = 4p... | 19 | https://mathoverflow.net/users/3503 | 19050 | 12,685 |
https://mathoverflow.net/questions/19046 | 40 | I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics).
I would likely use it for group theory, number theory, algebraic geometry and probably polytopes.
The kind of program... | https://mathoverflow.net/users/4619 | Open source mathematical software | Here are some links.
* [Axiom](http://www.axiom-developer.org/) and [Maxima](http://maxima.sourceforge.net/) are good general purpose computer algebra systems.
* [DataMelt](http://jwork.org/dmelt/) is a free Java-based math software with a lot of examples
* [GAP](http://www.gap-system.org/) is a system for computatio... | 65 | https://mathoverflow.net/users/nan | 19051 | 12,686 |
https://mathoverflow.net/questions/19054 | 2 | I'm looking for a formulas book.
I'm currently student in Communication Systems and we have several courses involving mainly complex analysis, fourier analysis, signal processing, information theory and sometimes other principles and I need a lot of books for all these formulas.
Does someone knows a book with these... | https://mathoverflow.net/users/4581 | Complete formulas book for Communication System engineer | I would suggest "Handbook of Formulas and Tables for Signal Processing", by Alexander D. Poularikas.
[This link](http://www.crcnetbase.com/isbn/978-0-8493-8579-7) should help you.
[This](http://rads.stackoverflow.com/amzn/click/0849385792) is the amazon link.
| 5 | https://mathoverflow.net/users/2938 | 19055 | 12,689 |
https://mathoverflow.net/questions/19070 | 15 | I am interested in learning the theory of types, especially in how they can provide a foundation to mathematics different to sets and how they can avoid self-referential paradoxes by stipulating that a collection of objects of type n has type n+1. As for my background knowledge, I only know a little of propositional an... | https://mathoverflow.net/users/4692 | Reference request for type theory | I would suggest you look at Martin-Löf's work, such as the following reprint of his earlier unpublished manuscript (from 1972?):
* Per Martin-L: [An Intuitionistic Theory of Types](http://books.google.si/books?id=pLnKggT_In4C&printsec=frontcover&hl=en&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false). In: Twenty-Fi... | 14 | https://mathoverflow.net/users/1176 | 19071 | 12,701 |
https://mathoverflow.net/questions/19066 | 12 | Is there a standard notion of non-degeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
| https://mathoverflow.net/users/1409 | Non-degenerate multilinear forms | I'm not sure if this notion is "standard", but there is one such notion, used for example in Nigel Hitchin's paper [*Stable forms and special metrics*](http://www.ams.org/mathscinet-getitem?mr=1871001) ([arXiv:math/0107101](http://arxiv.org/abs/math/0107101)) for alternating multilinear forms. The idea is that symplect... | 10 | https://mathoverflow.net/users/394 | 19073 | 12,702 |
https://mathoverflow.net/questions/19032 | 4 | Let E be a Hilbert C\*-module over some C\*-algebra and let $h \in K(E)$. Due to B. Blackadar's, "K-Theory for Operator algebras" Thm. 17.11.4 for a separable C\*-algebra $A$, represented by elements of $B(E)$, it is possible to construct a countable approximate unit $\{u\_n\}$ contained in $C^\*(h)$, such that $\{u\_n... | https://mathoverflow.net/users/4807 | Approximate unit for the algebra C*(h) consisting of projectors | No. In fact, K(E) need not even contain any nonzero projections. Take a (nontrivial) C\*-algebra B with no nonzero projections1 and take E=B as a right module over B with inner product 〈a,b〉=a\*b. Then K(E)≅B, as mentioned in [Example 13.2.4 (a)](http://books.google.com/books?id=_YQvFHnD6bQC&lpg=PA109&ots=w9jB2qq8x4&dq... | 3 | https://mathoverflow.net/users/1119 | 19075 | 12,703 |
https://mathoverflow.net/questions/19076 | 33 | Some MOers have been skeptic whether something like *natural number graphs* can be defined coherently such that every finite graph is isomorphic to such a graph. (See my previous questions [[1](https://mathoverflow.net/questions/17989/can-every-finite-graph-be-represented-by-one-prescribed-sequence-of-natural-numbe)], ... | https://mathoverflow.net/users/2672 | Bringing Number and Graph Theory Together: A Conjecture on Prime Numbers | **Theorem:** Schinzel's hypothesis H implies the conjecture.
**Proof:** Choose distinct primes $q\_S > 100|G|$ indexed by the 2-element subsets $S$ of $G$. For each $i \in G$, let $Q\_i$ be the set of $q\_S$ for $S$ such that $i \in S$ and the edge $S$ is not part of $G$. Let $P\_i$ be the product of the primes in $... | 40 | https://mathoverflow.net/users/2757 | 19080 | 12,706 |
https://mathoverflow.net/questions/17010 | 15 | I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^\*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It seems that later in Hopkins notes he says that the complex orientations of E are in one to one correspondence with multi... | https://mathoverflow.net/users/3901 | Complex orientations on homotopy | The natural starting point of this story are E-orientations on, say closed, manifolds M. That's just a *fundamental class* $[M^n] \in E\_n(M)$ such that cap product induces a (Poincare duality) isomorphism.
Given E, the question becomes which M are E-orientable. In many cases it happens that this follows if the stable... | 21 | https://mathoverflow.net/users/4625 | 19085 | 12,708 |
https://mathoverflow.net/questions/19084 | 1 | If $\mathcal M$ is a model category and I know that $A$ and $B$ are isomorphic in $\mathrm{Ho}(\mathcal M)$, is it guaranteed that there is a zig-zag of weak-equivalences in $\mathcal M$ connecting $A$ and $B$?
| https://mathoverflow.net/users/4466 | Equivalences in Model Categories | Yes. The isomorphism in $\mathrm{Ho}(\mathcal{M})$ is represented by a morphism in $\mathcal{M}$ from a cofibrant replacement for $A$ to a fibrant replacement for $B$. The "converse to the Whitehead lemma" states that a map in a model category is a weak equivalence iff its image in the homotopy category is an isomorphi... | 6 | https://mathoverflow.net/users/126667 | 19089 | 12,711 |
https://mathoverflow.net/questions/19091 | 1 | Let $X$ be a complex algebraic variety, and let $F=(F\_n)$ be a lisse $\mathbb Z\_{\ell}$-sheaf on $X.$ Does there exist an analytic open covering of $X(\mathbb C),$ such that $F$ is "(locally) constant" on each open subset?
| https://mathoverflow.net/users/370 | lisse sheaf on complex varieties | Dear Shenghao, If you really do mean a lisse sheaf on the etale site of $X$, then it doesn't make sense a priori to evaluate it on analytic open subsets of $X(\mathbb C)$, since these are not in the etale site of the algebraic variety $X$. However, $F$ corresponds to a representation of the (profinite) etale $\pi\_1$ o... | 7 | https://mathoverflow.net/users/2874 | 19093 | 12,712 |
https://mathoverflow.net/questions/19008 | 6 | My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\mathbb{N}$.
>
> Given a closed subscheme $X$ of $\mathbb{P}^n$ with Hilbert Polynomial $P$. The tangent space $T\_{[X... | https://mathoverflow.net/users/2625 | Question on a theorem of Eisenbud's and Harris' "The geometry of schemes" | Dear roger123,
This is largely a response to your question aksed as a comment below Charles Siegel's answer, but it won't fit in the comment box. Since $\mathcal N\_{\mathbb P^n/X}$ is a sheaf of modules over the sheaf of rings $\mathcal O\_{\mathbb P^n}$ (the structure sheaf of projective space),
its global section... | 4 | https://mathoverflow.net/users/2874 | 19094 | 12,713 |
https://mathoverflow.net/questions/19092 | 8 | I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the **Monthly**.
If so, **would you be so kind as to let me know about the main ideas in Beal's approach**? As you know, the whole point of his note is to present a solution to the following exercise in Her... | https://mathoverflow.net/users/1593 | On order of subgroups in abelian groups | The question is to prove that if $H$ and $K$ are subgroups of a
finite Abelian group or orders $m$ and $n$ then $G$ has a subgroup of order
$\mathrm{lcm}(m,n)$.
Beals starts by doing the case where $H$ and $K$ are cyclic. He proves
that $H$ is an internal direct product of cyclic groups of prime power orders.
Then he... | 9 | https://mathoverflow.net/users/4213 | 19096 | 12,715 |
https://mathoverflow.net/questions/19090 | 6 | The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both lengthwise and angular) between pairs of lines. There is also a converse statement. Suppose we are living in a complete Ri... | https://mathoverflow.net/users/nan | Generalizing cosine rule to symmetric spaces | There exist generalizations of the trigonometry laws to symmetric spaces. The following [article](http://arxiv.org/PS_cache/math-ph/pdf/9910/9910041v1.pdf) by Ortega and Santander works out the trigonometry laws for the case of real symmetric spaces of constant curvature and the following [one](http://arxiv.org/abs/mat... | 6 | https://mathoverflow.net/users/1059 | 19099 | 12,717 |
https://mathoverflow.net/questions/19074 | 5 | Let J be a closed interval of real numbers whose length is finite and positive. Let f be a real valued
function defined on J which has a continuous second derivative at all points of J.
QUESTION: If P1,P2,P3 are any three pairwise distinct and non-collinear points on the graph of f(J), does
there always exist at leas... | https://mathoverflow.net/users/4423 | Can one "soup-up" the LAW OF THE MEAN in the following way? | The answer is yes. Suppose the contrary and rescale the picture so that $r=1$. We may assume that $P\_1$ and $P\_3$ are the endpoints of the graph. There must be points on the graph that are outside the circle - otherwise the curvature at $P\_2$ is at least 1. WLOG assume that there are points below the circle. The low... | 4 | https://mathoverflow.net/users/4354 | 19101 | 12,718 |
https://mathoverflow.net/questions/19040 | 3 | This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot of questions, let me begin with an example.
Let $\pi:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be the morphism defined... | https://mathoverflow.net/users/4333 | On finite endomorphisms of $\mathbf{P}^r$ | The claimed computation is still wrong. Let $m \equiv r \mod n$, with $0 \leq r < n$. Then the right answer is that
$$\pi\_\* \mathcal{O}(m) = \mathcal{O}( \lfloor (m+1)/n \rfloor-1)^{\oplus(n-r-1)} \oplus \mathcal{O}( \lceil (m+1)/n \rceil-1)^{\oplus(r+1)}.$$
Let $S$ be the source $\mathbb{P}^1$ and $T$ the target.... | 5 | https://mathoverflow.net/users/297 | 19108 | 12,721 |
https://mathoverflow.net/questions/19105 | 1 | This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.
>
> Let $K$ be a field and $S=K[X\_0,\ldots,X\_n]$ a graded ring. Set $X=Proj(S)$ and let $M$ be a graded $S$-module. The functors $\Gamma\_\*$ definied by
> $$
> \Gamma\_\*(\mathcal{F})=\bigoplus\_... | https://mathoverflow.net/users/2625 | Question on an exercise in Hartshorne: Equivalence of categories | The homomorphisms in the category of sheaves are not sheaves themselves. The hom sheaves have the data of things that are only homomorphisms over open subsets. So if $Y,Z$ are coherent $\mathcal{O}\_X$-modules and you are looking for $\mathcal{O}\_X$-module homomorphisms, you don't actually get $\mathcal{H}om(X,Y)$, wh... | 4 | https://mathoverflow.net/users/622 | 19110 | 12,723 |
https://mathoverflow.net/questions/19104 | 5 | Is there a space such that it doesn't have any deformation but its space of first order infinitesimal deformations is non trivial?
| https://mathoverflow.net/users/4821 | Deformations for complex space germs | Such germs of spaces exist. See Section 7.6 of Ravi Vakil's paper *[Murphy's law in Algebraic Geometry](http://arxiv.org/abs/math/0411469)*.
| 5 | https://mathoverflow.net/users/297 | 19113 | 12,726 |
https://mathoverflow.net/questions/19119 | 8 | Let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves (of abelian groups) on a topological space $X$ such that $\mathcal{G}(U)$ is a subgroup of $\mathcal{F}(U)$ for every open set $U$ in $X$. The sheaf associated to the presheaf $P(\mathcal{F}/\mathcal{G})$ defined by
$$
U\mapsto \mathcal{F}(U)/\mathcal{G}(U)
$$
is calle... | https://mathoverflow.net/users/2625 | Why must one sheafify quotients of sheaves? | For presheaves (of sets or groups) we know what this particular (or any) colimit operation is: apply the operation objectiwise (for each $U$). Now the sheafification preserves colimits, hence we apply sheafification to a colimit cocone on presheaves to obtain a colimit cocone in sheaves. Doing sheafification to the pre... | 16 | https://mathoverflow.net/users/35833 | 19122 | 12,731 |
https://mathoverflow.net/questions/18766 | 26 | Let $\approx$ be the binary relation on the class of finitely generated groups
such that $G \approx H$ iff $G$ and $H$ have isomorphic (unlabeled nondirected)
Cayley graphs with respect to suitably chosen finite generating sets. Is $\approx$ an equivalence relation?
| https://mathoverflow.net/users/4706 | Cayley graphs of finitely generated groups | The answer is no, as expected. The following proof is "joint work" with L. Scheele.
Consider $G=\mathbb{Z}$, $K=D\_\infty$ and $H:=\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Then $G \approx K$ and $K\approx H$, but $G \not\approx H.$
Indeed, the Cayley graph associated to $\{-1,1\}$ for G and the Cayley graph associat... | 18 | https://mathoverflow.net/users/3380 | 19124 | 12,733 |
https://mathoverflow.net/questions/19132 | 5 | Maybe this is an elementary question.
Suppose that $U$ is a non-principal $\kappa$-complete ultrafilter on $\kappa$ and consider the standard ultrapower $M\cong \textrm{Ult}\_U(V)$ along with the corresponding elementary embedding $j\_U : V \rightarrow M$\,.
We know that if $U$ is in addition normal, then every ele... | https://mathoverflow.net/users/4826 | Representation of elements in ultrapowers | What you get is that if j:V to M is the ultrapower by any ultrafilter U on any set X, then every element of M has the form j(f)([id]). You can prove this by building an isomorphism from the ultrapower to the sets of this form. This way of thinking is also known as "seed theory".
**Theorem.** Suppose that j:V to M is... | 7 | https://mathoverflow.net/users/1946 | 19135 | 12,740 |
https://mathoverflow.net/questions/19142 | 1 | Let R be a commutative ring with 1.
An R-module K has the 'S' property if K/T = K implies that the submodule T is trivial.
By Fitting's Lemma any Noetherian module has the 'S' property. There exist non-Noetherian modules with this property. For example the infinite product of Z\_{2}xZ\_{3}xZ\_{5}x... running over a... | https://mathoverflow.net/users/4828 | Characterization of a certain class of modules-broader than Noetherian | These are called **Hopfian modules**. I didn't see any particularly exciting general characterization, but there are several special case characterizations (that show up easily in a google or mathscinet search). There are also several papers devoted to giving "interesting" examples.
An exercise in Lam's Lectures on M... | 5 | https://mathoverflow.net/users/3710 | 19144 | 12,746 |
https://mathoverflow.net/questions/19063 | 7 | Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 with itself and making the upper and lower halves adjoint.
The result I need is an extension of this to tensor products.... | https://mathoverflow.net/users/66 | Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules | I know a couple of ways to get a Shapovalov type form on a tensor product. The details of what I say depends on the exact conventions you use for quantum groups. I will follow Chari and Pressley's book.
The first method is to alter the adjoint slightly. If you choose a \* involution that is also a coalgebra automorp... | 6 | https://mathoverflow.net/users/1799 | 19145 | 12,747 |
https://mathoverflow.net/questions/19127 | 93 | I heard the following two questions recently from [Carl Mummert](http://users.marshall.edu/~mummertc/), who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory some traction on complete metric spaces, by considering the countable objects as st... | https://mathoverflow.net/users/1946 | Is there a dense subset of the real plane with all pairwise distances rational? | Let me answer Question 2.
Strong version: no. Consider $[0,1]$ with distance $d(x,y)=|x-y|^{1/3}$. There is no even a triple of points with rational distances - otherwise there would be a nonzero rational solution of $x^3+y^3=z^3$.
Weak version: yes. Let $(X,d)$ be the space in question. Construct sets $S\_1\subset... | 62 | https://mathoverflow.net/users/4354 | 19153 | 12,751 |
https://mathoverflow.net/questions/19118 | 9 | I guess this question only requires standard knowledge, but I'm a bit rusty with highest weight theory. I'm trying to catch up, but maybe I don't need the theory in full generality.
**Background**
Let $V$ be a Euclidean space of dimension $n$, and consider the representations of the group $G = O(V) \cong O(n)$. If ... | https://mathoverflow.net/users/828 | Decomposing a tensor product | For $SO(n)$ a calculation using LiE gives:
(using partition notation so $W$ is [2])
and assuming $n$ is not small
For $W\otimes W$, [4],[3,1],[2,2],[2],[1,1],[]
(all with multiplicity one)
and for $W\otimes W\otimes W$,
1.[6] 2.[5,1] 3.[4,2] 1.[3,3] 1.[4,1,1] 2.[3,2,1] 1.[2,2,2] 3.[4] 6.[3,1] 2.[2,2] 3... | 4 | https://mathoverflow.net/users/3992 | 19154 | 12,752 |
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