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https://mathoverflow.net/questions/7782 | 40 | Let $H$ be a group. Can we find an automorphism $\phi :H\rightarrow H$ which is not an inner automorphism, so that given any inclusion of groups $i:H\rightarrow G$ there is an automorphism $\Phi: G\rightarrow G$ that extends $\phi$, i.e. $\Phi\circ i=i\circ \phi$?
| https://mathoverflow.net/users/2224 | Are the inner automorphisms the only ones that extend to every overgroup? | The answer is that the inner automorphisms are indeed characterized by the property of the existence of extensions to larger groups containing the original group. I learned as much from
[this blog entry](http://mathreader.livejournal.com/22934.html) (in Russian). The reference is *Schupp, Paul E.*, [**A characterizatio... | 39 | https://mathoverflow.net/users/2106 | 7792 | 5,338 |
https://mathoverflow.net/questions/7794 | 9 | I am puzzled with the following problem:
Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every of the given numbers represents a shortest distance which belongs to a distinct pair of points?"
What is an efficient ... | https://mathoverflow.net/users/2266 | Feasibility of a list of prescribed distances in R^3 | Let us label the points $x\_0,x\_1,\dots,x\_m$.
For each pair $(x\_i,x\_j)$ prescribe the a value from your list;
denote it by say $d(x\_i,x\_j)$.
If there are no repititions then all together you have $N=[m{\cdot}(m+1)]!$ ways to do this.
Note that $d(x\_i,x\_j)$ completely determine "$m\times m$-matrix of scalar p... | 18 | https://mathoverflow.net/users/1441 | 7799 | 5,341 |
https://mathoverflow.net/questions/7627 | 17 | Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set theory? Is there any alternative foundation which is not a set-theory?
| https://mathoverflow.net/users/2266 | Set theory and alternative foundations | Bill Lawvere has suggested axiomatizing the category of categories as a foundation of mathematics, and there is no sense in which this could be thought of as a set theory. Colin McLarty is one person who has done some work on achieving such an axiomatization.
| 15 | https://mathoverflow.net/users/1106 | 7802 | 5,343 |
https://mathoverflow.net/questions/7800 | 7 | Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z\_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ over $R$, i.e., $R[G]$-modules of the form $R[G/H]$, where $H\subset G$ is a subgroup, and direct sums of such modules. Th... | https://mathoverflow.net/users/2106 | Exact sequences of permutational representations? | [Here is a preprint](http://math.ucsc.edu/~boltje/publications/p09y.pdf) by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S\_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.
(An earlier version of this answer had some out-of-step comments about resolutio... | 5 | https://mathoverflow.net/users/1450 | 7806 | 5,346 |
https://mathoverflow.net/questions/7808 | 5 | Suppose that I have a map of simplicial spaces,
$ f: X\_\* \to Y\_\*$,
and that I know that the map on zero spaces $f\_0: X\_0 \to Y\_0$ is n-connected. Can I conclude anything about the connectivity of the map of geometric realizations?
$ |f|: |X| \to |Y|$
Are there any reasonable conditions I can place on the... | https://mathoverflow.net/users/184 | Connectivity after Geometric Realization? | Yes it is a surjection on $\pi\_0$, because each component of $|Y|$ has at least one component of $Y\_0$.
Beyond that there are no restrictions. For instance, you can get any homotopy type for $|X|$ and $|Y|$ and any homotopy type for the map between them with $X\_0$ and $Y\_0$ just one point, as long as you ask that... | 4 | https://mathoverflow.net/users/1450 | 7809 | 5,348 |
https://mathoverflow.net/questions/7793 | 17 | My question is motivated by [Are the inner automorphisms the only ones that extend to every overgroup?](https://mathoverflow.net/questions/7782/are-the-inner-automorphisms-the-only-ones-that-extend-to-every-overgroup)
What are the auto-equivalences of the category of groups? What kind of structure do they form?
The... | https://mathoverflow.net/users/1657 | What are the auto-equivalences of the category of groups? | Suppose $F:\mathrm{Grp}\to\mathrm{Grp}$ is an equivalence. The object $\mathbb{Z}\in\mathrm{Grp}$ is a minimal generator (it is a generator, and no proper quotient is also a generator), and this property must be preserved by equivalences. Since there is a unique minimal generator, we can fix an isomorphism $\phi:\mathb... | 8 | https://mathoverflow.net/users/1409 | 7818 | 5,356 |
https://mathoverflow.net/questions/7817 | 8 | I'm hoping that the following are true. In fact, they are probably easy, but I'm not seeing the answers immediately.
Let $M$ be a smooth $m$-dimensional manifold with chosen positive smooth density $\mu$, i.e. a chosen (adjectives) volume form. (A *density* on $M$ is a section of a certain trivial line bundle. In loc... | https://mathoverflow.net/users/78 | Normal coordinates for a manifold with volume form | Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. ... | 9 | https://mathoverflow.net/users/1450 | 7819 | 5,357 |
https://mathoverflow.net/questions/22 | 47 | [Pablo Solis](http://math.berkeley.edu/~pablo/) asked this at a recent [20 questions seminar](http://scratchpad.wikia.com/wiki/20qs) at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very unlikely, but I don't have a proof. It's easy to s... | https://mathoverflow.net/users/3 | Can $N^2$ have only digits 0 and 1, other than $N=10^k$? | In the interest of completeness, here is what I put on the 20-questions wiki — we might as well repeat it here in the $\infty$-questions site. I had basically the same idea as Ilya, do a branched search to look for the digits of $N^2$. However, the code that I wrote in Python works from the 10-adic end, while Ilya's wo... | 16 | https://mathoverflow.net/users/1450 | 7825 | 5,361 |
https://mathoverflow.net/questions/7828 | 9 | A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.
What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $l^3$-norm might be induced by a trilinear form in a similar fashion like the $l^2$-norm by a bilinear form is.
Is the... | https://mathoverflow.net/users/2082 | Hilbert spaces are induced by a bilinear form. How about n-linear forms? | As long as the form is positive definite and the unit ball is convex, you get a perfectly good Banach space using any symmetric $n$-linear form on a real vector space $V$. The degree $n$ is necessarily even. It is equivalent to defining the norm as the $n$th root of a homogeneous degree $n$ polynomial. $\ell^p$ is an e... | 8 | https://mathoverflow.net/users/1450 | 7832 | 5,366 |
https://mathoverflow.net/questions/7836 | 55 | I have this question coming from an earlier [Qiaochu's post](https://mathoverflow.net/questions/5243/why-is-it-a-good-idea-to-study-a-ring-by-studying-its-modules). Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is ther... | https://mathoverflow.net/users/nan | Why is it useful to study vector bundles? | Well, in algebraic geometry, here's a couple of reasons:
1) Subvarieties: Take a vector bundle, look at a section, where is it zero? Lots of subvarieties show up this way (not all, see [this question](https://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle)) but generally, ... | 33 | https://mathoverflow.net/users/622 | 7837 | 5,368 |
https://mathoverflow.net/questions/7824 | 6 | Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of whether something is natural or not, studying it extends our knowledge of mathematics, which is always good. But the uni... | https://mathoverflow.net/users/1916 | Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements? | This is sort of an anti-answer, but: my instinct is that ZC is taking the categorical perspective too far.
To start philosophically, I think it is quite appropriate to, when given a mathematical structure like a topological space or a ring -- i.e., a set with additional structure -- refrain from inquiring as to exac... | 3 | https://mathoverflow.net/users/1149 | 7839 | 5,370 |
https://mathoverflow.net/questions/7823 | 14 | I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to the count rather than $1$.
However, I don't understand why or how Gromov-Witten invariants can be negative. What is t... | https://mathoverflow.net/users/83 | Negative Gromov-Witten invariants | Gromov--Witten invariants are designed to count the "number" of curves in a space in a deformation invariant way. Since the number of curves can change under deformations, the Gromov--Witten invariants won't have a direct interpretation in terms of actual numbers of curves, even taking automorphisms into account.
Her... | 14 | https://mathoverflow.net/users/32 | 7841 | 5,371 |
https://mathoverflow.net/questions/7840 | 9 | What is an example of a finite local rings, that has length 2 or 3?
I want something different from $F\_{q}[x] / x^{i}$ for $i=2, 3$; I'm looking for something more interesting. If you can give me examples of higher length, yet have "simple structure" (e.g. $F\_{q}[x]/x^{i}$), that would be nice too.
I know this is... | https://mathoverflow.net/users/2623 | Examples of finite local rings of length 2 or 3 | There are a bunch of different notions of length/depth in ring theory: Projective length, Artinian length, local depth, etc. If we take length to mean Artinian length, then Charles is right: The Artinian length of a finite-dimensional commutative algebra is just its dimension. Every such algebra is a direct sum of loca... | 5 | https://mathoverflow.net/users/1450 | 7843 | 5,373 |
https://mathoverflow.net/questions/7859 | 0 | I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).
She understood solving equations of the form $ax + b = 0$ by first adding $-b$ to both sides and then diving by $a$. Dealing with ne... | https://mathoverflow.net/users/1445 | How to teach addition of negative numbers? | It would help to know a little more about the nature of your friend's dyscalclia. The sense of the integer line comes from the inferior parietal cortex (and causes difficulty with problems like "what number is halfway between 7 and 11?") while rote memory-type problems ("what is 4 times 7?") are associated with the bas... | 15 | https://mathoverflow.net/users/441 | 7866 | 5,387 |
https://mathoverflow.net/questions/7847 | 3 | The following is from this talk: <http://www.maths.usyd.edu.au/u/anthonyh/piecestalk.pdf>, Slide 14.
The Springer correspondence gives bijections
SO2n+1 \ N(so2n+1) ↔ {(μ; ν) | μi ≥ νi − 2, νi ≥ μi+1},
Sp2n \ N(sp2n) ↔ {(μ; ν) | μi ≥ νi − 1, νi ≥ μi+1 − 1},
obtained from the previous parametrizations by taking... | https://mathoverflow.net/users/2623 | A bijection between "symplectic" partitions and bi-partitions via Springer correspondance | From looking at the slides, it sure looks like you wrote it in your answer: take 2-quotients.
I'm not sure if there's a standard reference for n-quotients of partitions, but they're described in [this paper](http://arxiv.org/abs/math.CO/0609175), for example.
| 2 | https://mathoverflow.net/users/66 | 7871 | 5,391 |
https://mathoverflow.net/questions/7858 | 2 | What is the standard resolution of singularities, for the nilpotent cone (of the adjoint representation) for the symplectic group? I know how to do this for the general linear group, but am having trouble finding a good reference for the symplectic group. I understand it uses Richardson orbits.
(I do know what the clo... | https://mathoverflow.net/users/2623 | Resolution of singularities for nilpotent cone of the symplectic group | Ben gave the general answer above. If you care specifically about the symplectic group and are interested in a "flag-like" description of its flag variety, then one exists. It is given by all half-flags of isotropic subspaces (this is just like for $SL\_n$, the symplectic group acts transitively and the stabilizer of t... | 2 | https://mathoverflow.net/users/916 | 7874 | 5,394 |
https://mathoverflow.net/questions/7881 | 16 | Let $n,k$ be positive integers. What is the smallest value of $N$ such that for any $N$ vectors (may be repeated) in $(\mathbb Z/(n))^k$, one can pick $n$ vectors whose sum is $0$?
My guess is $N=2^k(n-1)+1$. It is certainly sharp: one can pick our set to be $n-1$ copies of the set $(a\_1,...,a\_k)$, with each $a\_i... | https://mathoverflow.net/users/2083 | Sum of $n$ vectors in $(\mathbb Z/n)^k$ | Your guess is correct for k=1 and 2, but when k is bigger, things get more complicated. For instance, when k=n=3, N=19. For a summary of some known results, see:
[Elsholtz, C. Lower Bounds For Multidimensional Zero Sums. *Combinatorica* **24**, 351–358 (2004).](https://doi.org/10.1007/s00493-004-0022-y)
| 11 | https://mathoverflow.net/users/1013 | 7890 | 5,408 |
https://mathoverflow.net/questions/7892 | 14 | Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL manifolds, this is equivalent to whether the space of smooth structures on a PL 4-manifold is contractible. (I think it's known... | https://mathoverflow.net/users/2327 | Smooth structures on PL 4-manifolds | Very little is known about that question, the same smoothing theory gives something that I'm trying to get people to call "The Cerf-Morlet Comparison Theorem"
$$ Diff(D^n) \simeq \Omega^{n+1}(PL(n)/O(n)) $$
$Diff(D^n)$ is the group of diffeomorphisms of the $n$-ball where the diffeomorphisms are pointwise fixed on ... | 10 | https://mathoverflow.net/users/1465 | 7894 | 5,410 |
https://mathoverflow.net/questions/4454 | 29 | Is there a *natural* measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or not? By the word 'natural' I am trying to exclude measures defined in terms of the characteristic function of the set of... | https://mathoverflow.net/users/1508 | How many of the true sentences are provable? | It seems to me that the probability that a statement is provable and that it is undecidable should both be bounded away from 0, for any reasonable probability distribution.
Let $C\_n$ be the number of grammatical statements of length $n$. For any statement $S$, the statement
>
> $S$, or $1=1$
>
>
>
is a theo... | 31 | https://mathoverflow.net/users/297 | 7902 | 5,415 |
https://mathoverflow.net/questions/7897 | 11 | I heard people mentioned this in one sentence, but don't see the reason.
Why a (smooth) variety of general type, i.e. an algebraic variety X with K\_X big, is hyperbolic, i.e. has no non-constant map from the complex number into it?
I don't know what are the necessary assumption on the variety, do we need propernes... | https://mathoverflow.net/users/1657 | Why is a variety of general type hyperbolic? | You must be thinking of Lang's conjecture which predicts that a smooth projective variety is (Brody) hyperbolic if and only if all of its irreducible subvarieties are of general type.
This is still not known in general but there are many special cases that are known. A good example is McQuillan's theorem - a smooth s... | 9 | https://mathoverflow.net/users/439 | 7905 | 5,416 |
https://mathoverflow.net/questions/7911 | 19 | I am looking for a counter-example of two functors F : C -> D and G : D->C such that
1) F is left adjoint to G
2) F is right adjoint to G
3) F is not an equivalence (ie F is not a quasi-inverse of G)
| https://mathoverflow.net/users/2330 | Is a functor which has a left adjoint which is also its right adjoint an equivalence ? | The answer of Ben Webster, can be made easier. Consider the functor F : (A-mod) -> (A-mod) which maps any A-module on (0). Then, F is a left adjoint to F ; and so, is a also a right adjoint to F. This is clear because for all A-modules N, M, one has Hom\_A(0,N)=Hom\_A(M,0). But, F is not an equivalence.
| 13 | https://mathoverflow.net/users/2330 | 7922 | 5,427 |
https://mathoverflow.net/questions/7926 | 2 | I´m looking for information about the intersection of two vector bundles (principally trivial bundles, but no necessarily). I´m trying to make a picture (literally) of reflexive finite generated modules.
Another related topic is sub-budles of a vector bundles.
All suggestions are wellcome!
---
**Edit*... | https://mathoverflow.net/users/2040 | About the intersection of two vector bundles | You need to refine the question to get better answers, but here are some thoughts:
1) Over a normal variety, you can think of line bundles as divisors, and "intersect" them.
2) A vector bundle can be represented by a reflexive sheaf, but being reflexive is a lot weaker.
| 3 | https://mathoverflow.net/users/2083 | 7932 | 5,433 |
https://mathoverflow.net/questions/7942 | 2 | Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
* rank A=1 ↔ all 2 x 2 minors are =0.
* J:= ideal generated by a i j verify (R:(R:J))=R ↔ J is not included in any prime ideal with heig... | https://mathoverflow.net/users/2040 | Splitting matrix of rank one | I am having trouble understanding your English. But, if I understand you correctly, the following is a counter-example:
Let $k$ be a field and let $R$ be the ring $k[a,b,c,d]/(ab-cd)$. Then $R$ is normal and $\left( \begin{smallmatrix} a & c \\\\ d & b \end{smallmatrix} \right)$ has rank 1. However, we can not write ... | 3 | https://mathoverflow.net/users/297 | 7945 | 5,440 |
https://mathoverflow.net/questions/7951 | 4 | Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that it satisfies the descending chain condition), and P has countably infinite automorphism group?
The question is motiva... | https://mathoverflow.net/users/382 | Is there a poset with 0 with countable automorphism group? | It seems unlikely (once you assume d.c.c.). Define the height of an element $x$ in $P$ to be the length of the shortest unrefinable chain from $x$ to $0$.
Let $P\_n$ denote the elements of $P$ whose height is at most $n$. Since each element has a finite number of covers, the number of elements in $P\_n$ is finite.
... | 5 | https://mathoverflow.net/users/468 | 7955 | 5,447 |
https://mathoverflow.net/questions/7907 | 10 | How can I determine the integer points of a given elliptic curve if I know its rank and its torsion group?
I read same basic books on elliptic curves but as a non-professional I didn't understand everything. Is it true that if rank is 0 and torsion group is isomorphic to a group of order $n$ then the number of integ... | https://mathoverflow.net/users/2120 | How to find all integer points on an elliptic curve? | Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical software that does it for you (e.g. MAGMA), and then let it run until it either finds the answer or runs out of memory. Al... | 9 | https://mathoverflow.net/users/1384 | 7979 | 5,460 |
https://mathoverflow.net/questions/7870 | 0 | How does one linearize and analyze such a system?
Just noticed I could edit this, so from my comment below:
I am trying to get a feel for what analysis us used beyond the introduction I have had. The equations for the double pendulum were derived from the second derivative of the equations for position of each mass... | https://mathoverflow.net/users/2320 | Stability analysis of a system of 2 second order nonlinear differential equations | This is an answer to Charles' restatement of the question.
Recall that equation F(x,x',x'') = 0 (e.g. x'' + sin x = 0) can be written as a system
X' = f(X), where X = (x,x')^T (e.g. f(x',x) = (-sin x, x')^T) and that system can be linearized about an equilibrium E = (x\_*,x'\_*)^T to obtain a linear equation
X' = A... | 3 | https://mathoverflow.net/users/1281 | 7993 | 5,469 |
https://mathoverflow.net/questions/7968 | 2 | Let $N\_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N\_3$? I don't see how to extend the ideas related to the 3d lens spaces. Any feedback would be super-welcome
| https://mathoverflow.net/users/2196 | Two solid N_3 glued by its boundary | It is a general fact that a closed manifold of odd Euler characteristic cannot bound a compact manifold. This can be deduced pretty easily from the fact that a closed manifold of odd dimension has Euler characteristic zero (a consequence of Poincaré duality) as follows. Suppose N is the boundary of a compact manifold P... | 13 | https://mathoverflow.net/users/23571 | 8005 | 5,478 |
https://mathoverflow.net/questions/8014 | 5 | I'm a bit stuck, and I'm hoping someone can help me out. I have a vector bundle $E$ on an algebraic curve (the ones I am interested in are holomorphic, but I'm sure that doesn't matter so much for the purposes of this question...), and I also have an endomorphism $\Psi\in\Gamma(\mbox{End}(E)).$
How **in general** doe... | https://mathoverflow.net/users/2347 | Endomorphisms of vector bundles | The determinant of $\Psi$ is the top exterior power $\bigwedge^{\mathrm{rk}(E)}\Psi$ (identifying endomorphisms of the line bundle $\bigwedge^{\mathrm{rk}(E)}E$ with functions).
| 10 | https://mathoverflow.net/users/2035 | 8018 | 5,487 |
https://mathoverflow.net/questions/8023 | 13 | I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for $n=1$, all Riemannian manifolds are modeled on $\mathbb R$. When $n=2$, I believe that it suffices for the scalar curvat... | https://mathoverflow.net/users/78 | When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$? | If the Riemannian metric is twice differentiable in some co-ordinate system, then this holds in any dimension if and only if the Riemann curvature tensor vanishes identically.
In dimension 2, it suffices for the scalar curvature to vanish.
In dimension 3, it suffices for the Ricci curvature to vanish.
In higher dimen... | 19 | https://mathoverflow.net/users/613 | 8026 | 5,493 |
https://mathoverflow.net/questions/7998 | 16 | Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them?
I'm wondering because I find it difficult to come up with a lot of examples of nontrivial pairwise independent random variables. (by 'nontrivial', i mean not mutually independent)
one example (three ... | https://mathoverflow.net/users/1354 | most general way to generate pairwise independent random variables? | I'm sure that Gil's answer is wise and that it is a good idea to look at Alon and Spencer's book. Here also is a quick summary of what is going on.
Suppose that $X\_1,\ldots,X\_n$ are random variables, and suppose for simplicity that they take finitely many values. Suppose that you prescribe the distribution of each ... | 7 | https://mathoverflow.net/users/1450 | 8027 | 5,494 |
https://mathoverflow.net/questions/8003 | 19 | The following is a theorem of [Elkies](http://www.ams.org/mathscinet-getitem?mr=903384):
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
Allen Knutson and I would like a similar theorem for higher dimension... | https://mathoverflow.net/users/297 | Elkies' supersingularity theorem in higher dimension | If I understand your question correctly, the "easy" version of the question you ask is unknown in dimension $\ge 3$, and is basically easy for surfaces of interest.
These questions are certainly unknown in dimension $\ge 3$, or else the Sato-Tate conjecture for weight $>3$ would have been known $12$ months ago, rathe... | 17 | https://mathoverflow.net/users/nan | 8035 | 5,500 |
https://mathoverflow.net/questions/8031 | 12 | The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a circle packing whose graph is G. What happens when circles are replaced by spheres? By spheres of higher dimension? What is known about the graphs that are forme... | https://mathoverflow.net/users/1098 | Graphs of Tangent Spheres | The number of edges in such a graph is linear in the number of vertices, and they can be split into two equal-sized subgraphs by the removal of $O(n^{\frac{d-1}{d}})$ vertices. See e.g.
A deterministic linear time algorithm for geometric separators and its applications.
D. Eppstein, G.L. Miller, and S.-H. Teng.
[Fund... | 13 | https://mathoverflow.net/users/440 | 8036 | 5,501 |
https://mathoverflow.net/questions/7938 | 4 | How do you compute in characteristic $0$, intersection cohomology of partial flag varieties (corresponding to a fixed partition $\lambda$)? I understand the answer involves Kazhdan-Lusztig polynomials; all I can find is a reference for characteristic $p$ (<http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0207v2.pdf>), I'm... | https://mathoverflow.net/users/2623 | Intersection cohomology of flag varieties/Schubert varieties | First, let me rephrase your question in a slightly pedantic manner.
To establish some notation, for a point $p$ on the flag variety $G/B$, let $V\_1(p)\subset\cdots V\_{n-1}(p)$ be the flag in $\mathbb{C}^n$ that it corresponds to. (Be careful. There are no flags actually in the flag variety, just points. Rather, the... | 3 | https://mathoverflow.net/users/3077 | 8047 | 5,508 |
https://mathoverflow.net/questions/8039 | 23 | This question comes out of the answers to [Ho Chung Siu's](https://mathoverflow.net/questions/7836/why-is-it-useful-to-study-vector-bundles) question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of generality, starting with vector bundles and ... | https://mathoverflow.net/users/290 | What is a section? | **To your first question,** "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc. (This is a fairly selective use of the word "function" which used to confuse me... | 25 | https://mathoverflow.net/users/84526 | 8049 | 5,510 |
https://mathoverflow.net/questions/8038 | 7 | Chebyshev polynomials have real roots and satisfy a recurrence relation. I was wondering if one can find a sequence of polynomials with integral or rational roots with similar properties. More precisely, one is looking for a sequence of polynomials $(f\_n),f\_n\in\mathbf{Q}[t]$ such that
1. $\deg f\_n\to\infty$ as $n... | https://mathoverflow.net/users/2349 | Chebyshev-like polynomials with integral roots | Here's one thought. For each integer k, f\_n(k) satisfies a recurrence relation. If the roots of f\_n are all integers, then f\_n(k) and f\_n(k+1) have to be "in sync" in the sense that they never have opposite sign. This is a strong condition! For instance, suppose the sequences f\_n(k) and f\_n(k+1) each have unique ... | 3 | https://mathoverflow.net/users/431 | 8051 | 5,512 |
https://mathoverflow.net/questions/7921 | 27 | Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is there an explicit compact counterexample, i.e., are there two compact smooth 4-manifolds which are homeomorphic, have isomor... | https://mathoverflow.net/users/2327 | Failure of smoothing theory for topological 4-manifolds | For a pair of smooth, simply connected, compact, oriented 4-manifolds $X$ and $Y$,
* Any isomorphism of the intersection lattices $H^2(X)\to H^2(Y)$ comes from an oriented homotopy equivalence $Y\to X$ (Milnor, 1958).
* Any oriented homotopy equivalence is a tangential homotopy equivalence (Milnor, Hirzebruch-Hopf 19... | 24 | https://mathoverflow.net/users/2356 | 8054 | 5,514 |
https://mathoverflow.net/questions/8032 | 9 | It seems a well-known fact that subvarieties of a variety of general type containing a general point are also of general type. This fact is an essential property used to prove some extension theorems of pluricanonical forms on algebraic varieties of general type, see for example the very nice survey on extension of plu... | https://mathoverflow.net/users/2348 | Why a subvariety of a variety of general type is of general type | Let me reproduce the relevant bit from the [reference](http://www.iecl.univ-lorraine.fr/~Gianluca.Pacienza/notes-grenoble.pdf) above ([Internet Archive](http://web.archive.org/web/20190320105238/http://www.iecl.univ-lorraine.fr/~Gianluca.Pacienza/notes-grenoble.pdf)):
>
> **Exercise 3.1.** Let $X$ be a variety of g... | 4 | https://mathoverflow.net/users/605 | 8055 | 5,515 |
https://mathoverflow.net/questions/8052 | 56 | I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with it? In other words, if one were to try to do everything without spectral sequences and only using more elementary argumen... | https://mathoverflow.net/users/344 | Why are spectral sequences so ubiquitous? | 1. Let's say you have a resolution $0\to A\to J^0\to J^1\to\dots$ (of a module, a sheaf, etc.) If $J^n$ are acyclic (meaning, have trivial higher cohomology, resp. derived functors $R^nF$), you can use this resolution to compute the cohomologies of $A$ (resp. $R^nF(A)$). If $J^n$ are not acyclic, you get a spectral seq... | 44 | https://mathoverflow.net/users/1784 | 8057 | 5,516 |
https://mathoverflow.net/questions/8042 | 32 | The title pretty much says it all: suppose $R$ is a commutative integral domain such that every countably generated ideal is principal. Must $R$ be a principal ideal domain?
More generally: for which pairs of cardinals $\alpha < \beta$ is it the case that: for any commutative domain, if every ideal with a generating ... | https://mathoverflow.net/users/1149 | Do there exist non-PIDs in which every countably generated ideal is principal? | No such ring exists.
Suppose otherwise. Let $I$ be a non-principal ideal, generated by a collection of elements $f\_\alpha$ indexed by the set of ordinals $\alpha<\gamma$ for some $\gamma$. Consider the set $S$ of ordinals $\beta$ with the property that the ideal generated by $f\_\alpha$ with $\alpha<\beta$ is not e... | 38 | https://mathoverflow.net/users/468 | 8067 | 5,524 |
https://mathoverflow.net/questions/8088 | 0 | For P a partially ordered set, let S be a subset of P such that if:
a,c\in S and b\in P and a<=b<=c then b\in S
Is there a name for a subset with this property? The term "dense" subset is already taken and means something else.
| https://mathoverflow.net/users/2361 | name for "solid" subset of a partially ordered set? | A set with this property is called *convex*.
See e.g. Quasi-uniform spaces, Volume 77 of Lecture notes in pure and applied mathematics, Peter Fletcher, William F. Lindgren, Marcel Dekker, 1982, p.84.
| 11 | https://mathoverflow.net/users/440 | 8089 | 5,536 |
https://mathoverflow.net/questions/8095 | 2 | I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ideal $B$ such that $A = B +C$ for any non-trivial two-sided ideal $C$ of $A$. I've tried reasoning it out, and I'm not s... | https://mathoverflow.net/users/2623 | Some equivalent statements about primitive algebras | Lam (A first course in noncommutative rings, 2ed) does it for (unital) rings $R$ in Lemma 11.28 (page 186):
>
> If such a $B$ exists, we may assume (after an application of Zorn's Lemma) that it is a maximal left ideal. The annihilator of the simple left $R$-module $R/B$ is an ideal in $B$, and so it must be zero. ... | 3 | https://mathoverflow.net/users/1234 | 8098 | 5,541 |
https://mathoverflow.net/questions/8097 | 16 | Most of the number theory textbooks I've dealt with take a very classical approach to the subject. I'm looking for a textbook that's something like a first course in number theory for people who have a decent command of modern algebra (at the level of something like Lang's Algebra). Does such a book exist, and if it do... | https://mathoverflow.net/users/1353 | Number theory textbook with an algebraic perspective | There are probably many such books, for instance "[Fundamentals of Number Theory](http://store.doverpublications.com/0486689069.html)" by LeVeque, "[Elementary Number Theory](http://store.doverpublications.com/0486458075.html)" by Bolker and "[A Classical Introduction to Modern Number Theory](http://www.springer.com/ma... | 10 | https://mathoverflow.net/users/532 | 8099 | 5,542 |
https://mathoverflow.net/questions/4790 | 15 | So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if you're willing to weaken to quasi-Hopf algebra, maybe this is all of them?).
I'm interested in the categorified version... | https://mathoverflow.net/users/66 | What structure on a monoidal category would make its 2-category of module categories monoidal and braided? | Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a (monoidal) *functor* $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak)... | 8 | https://mathoverflow.net/users/184 | 8103 | 5,546 |
https://mathoverflow.net/questions/8091 | 95 | You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope has the magical property that if you ever touch its ends to each other, they will stick together and never come apart for... | https://mathoverflow.net/users/2363 | Is it possible to capture a sphere in a knot? | Without loss of generality we can assume that rope is everywhere tangent to the sphere.
Then some infinitesimal Möbius tranform of its surface will shorten the wrapping length while preserving the crossing pattern (and therefore will loosen the rope without causing it to pass through itself).
Once it is proved, moving ... | 63 | https://mathoverflow.net/users/440 | 8111 | 5,553 |
https://mathoverflow.net/questions/8113 | 7 | What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}\_{2}$ /$\Pi^{0}\_{2}$?
| https://mathoverflow.net/users/158 | Higher-rank Borel sets | The Borel hierarchy is, of course, strictly increasing at
every step, so there will be sets of every countable
ordinal rank. Furthermore, all of these sets are relatively
concrete, obtained as countable unions of sets having lower
rank. But you asked for natural examples, so let me give a
few.
1) The collection of tr... | 5 | https://mathoverflow.net/users/1946 | 8121 | 5,560 |
https://mathoverflow.net/questions/8110 | 4 | I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U\_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody algebra $\mathfrak{g}$. Its negative part $U^{-}$ (the subalgebra of $U$ generated by the $f\_i$'s) has a canonical ba... | https://mathoverflow.net/users/914 | Canonical basis for the extended quantum enveloping algebras | The basis $\dot{B}$ is usually hard to compute, in the same way the Kazhdan-Lusztig basis for the Hecke algebra is. On the other hand, at least for type A, there is another way to get at this basis via what you might call "quantum Schur-Weyl duality": there's a geometric construction of finite dimensional quotients of ... | 6 | https://mathoverflow.net/users/1878 | 8122 | 5,561 |
https://mathoverflow.net/questions/8115 | 11 | Let $T\_1$ and $T\_2$ be two Grothendieck topologies on the same small category $C$, and let $T\_3 = T\_1 \cup T\_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T\_1$ and $T\_2$). If $X$ is a $T\_1$-sheaf, is its $T\_2$-sheafification still a $T\_1$-sheaf (and therefore a $T\_3$-sheaf)?
If ... | https://mathoverflow.net/users/49 | Does sheafification preserve sheaves for a different topology? | I think my answer to [this question](https://mathoverflow.net/questions/6592/localizing-model-structures) provides a counterexample: Let C be the category a → b, and consider the topologies T1 generated by the single covering family {a → b} and T2 generated by declaring the empty family to be a covering of a. (Caution:... | 10 | https://mathoverflow.net/users/126667 | 8123 | 5,562 |
https://mathoverflow.net/questions/8134 | 6 | It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." In which other categories is this also true? In particular is it true for abelian categories?
| https://mathoverflow.net/users/2376 | Existence of projective resolutions in abelian categories | Among the standard examples of abelian categories without enough projectives, there are
1. the categories of sheaves of abelian groups on a topological space (as VA said), or sheaves of modules over a ringed space, or quasi-coherent sheaves on a non-affine scheme;
2. the categories of comodules over a coalgebra or co... | 18 | https://mathoverflow.net/users/2106 | 8139 | 5,575 |
https://mathoverflow.net/questions/8137 | 4 | We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we have the length functions of the geodesics boundaries and the twisting parameters for gluing these boundaries.
My ques... | https://mathoverflow.net/users/2380 | Coordinates on Teichmuller space | You are describing the "grafting construction". I am not an expert: however if you google "grafting a Riemann surface" there are many references available.
If you take $h$ to be non-negative you cannot reach all of Teichmuller space in the way you describe. This is because the cuffs of your pair of pants decompositi... | 4 | https://mathoverflow.net/users/1650 | 8140 | 5,576 |
https://mathoverflow.net/questions/7200 | 2 | What results are known about representations of reductive groups over finite rings in general? Here by finite rings I usually mean an algebra over $F\_q$, I guess.
I know Lusztig has a paper generalizing Deligne-Lusztig theory to finite commutative rings of the form $F\_q[x]/x^{\epsilon}$ for some integer $\epsilon ... | https://mathoverflow.net/users/2623 | Representations of reductive groups over finite rings | Lusztig's results mentioned above have been generalised to groups over arbitrary finite local rings, see *Unramified representations of reductive groups over finite rings*, Represent. Theory 13 (2009), 636-656.
There is a recent paper extending the above construction to certain ramified maximal tori, cf. *Extended De... | 2 | https://mathoverflow.net/users/2381 | 8141 | 5,577 |
https://mathoverflow.net/questions/8145 | 16 | A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece.
Are there stick knots which are topologically trival, but not trivial via a stick isotopy?
| https://mathoverflow.net/users/3 | Are there piecewise-linear unknots that are not metrically unknottable? | Yes, there are. See "Locked and Unlocked Polygonal Chains in 3D", T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, S. Whitesides, [arXiv:cs.CG/9910009](http://arxiv.org/abs/cs.CG/9910009), figure 6.
| 22 | https://mathoverflow.net/users/440 | 8146 | 5,581 |
https://mathoverflow.net/questions/8147 | 1 | I´m looking for homomorphisms between exterior powers of a **free** module M of rank m
ΛmR M → Λm-1R M
Exactly, I´m looking for an **explicit** isomorphism
M → Hom R (ΛmR M , Λm-1R M)
I compare the ranks and the things go, but I can not imagine a concrete expression.
Suggestions are welcome
| https://mathoverflow.net/users/2040 | Homomorphism between exterior powers of a free module of finite rank | If $R$ is commutative, let $e\_1, \ldots, e\_m$ be an $R$-basis of $M$. Then $\Lambda^m M$ is a free $R$-module of rank one generated by the vector $e\_1 \wedge e\_2 \wedge \ldots \wedge e\_m$ and $\Lambda^{m-1}M$ is free of rank $m$ with basis $$\lbrace e\_1 \wedge \ldots \wedge \hat{e\_i} \wedge \ldots \wedge e\_m \r... | 6 | https://mathoverflow.net/users/1797 | 8149 | 5,583 |
https://mathoverflow.net/questions/7316 | 4 | I am now learning induction problems in representation theory. I know David Vogan's book cohomological induction and unitary representation theory might be good references,but it is too thick.
I wonder whether there is some good and detailed notes on using derived functor to construct irreducible representations. It... | https://mathoverflow.net/users/1851 | On the materials about cohomological induction | You might look at Vogan's orange book "Unitary Representations of Reductive Lie Groups," which is a much thinner book, or (thinner yet) chapters 5-6 of "Dirac Operators in Representation Theory," by Jing-Song Huang and Pavle Pandzic. These both have descriptions of cohomological induction, though off the top of my head... | 4 | https://mathoverflow.net/users/1322 | 8154 | 5,588 |
https://mathoverflow.net/questions/8075 | 6 | I've been reading about the Langlands program (the paper by Torsten Wedhorn "Local langlands correspondence for GL(n) over p-adic fields, to be precise), and I want to get my hands dirty with examples.
What are some interesting (yet not too nasty) examples of admissible representations of $GL\_{n}(K)$, where $K$ is a $... | https://mathoverflow.net/users/2623 | examples of admissible representations of $GL_{n}(K)$ over p-adic field | I second L Spice's recommendation of the book by Bushnell and Henniart, called "The local Langlands conjectures for GL(2)."
After you master the principal series representations, it's not too hard to tinker with some supercuspidals. Easiest among these are the tamely ramified supercuspidals. To construct these, let'... | 7 | https://mathoverflow.net/users/271 | 8163 | 5,594 |
https://mathoverflow.net/questions/8167 | 5 | If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course nth roots play a vital role in field theory, e.g. in the characterization of solvable extensions in characteristic 0. ... | https://mathoverflow.net/users/1149 | Notation/name for "Artin-Schreier roots"? | Google Scholar finds three papers with the phrase "Artin-Schreier root" or "Artin-Schreier roots" (with quotes). The papers are by Jing Yu, Thomas Scanlon, and Spencer Bloch + Helene Esnault. This is not all that many, but maybe enough for some kind of standard. Various people, probably enough to call it standard, also... | 3 | https://mathoverflow.net/users/1450 | 8169 | 5,598 |
https://mathoverflow.net/questions/8160 | 3 | Let $A$ be the set of all quadruples $(a\_0,a\_1,a\_2,a\_3) \in {\mathbb Q}^4$ such that
the polynomial $P=X^4+a\_3X^3+a\_2X^2+a\_1X+a\_0$ is irreducible and if $z$ is any root
of $P$, then ${\mathbb Q}(z)$ contains $\sqrt{2}$. Is there a nontrivial polynomial relation
$R(a\_0,a\_1,a\_2,a\_3)=0$ satisified by all $(a\_... | https://mathoverflow.net/users/2389 | Expressing field inclusions by polynomial equalities on coefficients | If there was a nontrivial polynomial relation between the coefficients, it would be true for a dense subset (~~reducibility is a nowhere dense condition~~ see comment below) of all polynomials of the form $(x^2+(\alpha +\beta\sqrt{2})x+\gamma+\delta\sqrt{2})(x^2+(\alpha -\beta\sqrt{2})x+\gamma-\delta\sqrt{2})$ with rat... | 1 | https://mathoverflow.net/users/2368 | 8171 | 5,600 |
https://mathoverflow.net/questions/8189 | 5 | What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy Sequences that are equivalent to any single real number.
| https://mathoverflow.net/users/2394 | Cardinality of Equivalence Classes of Cauchy Sequences | It's the same as the size of the real numbers. Here's a rough sketch of the proof.
For each element of (0,1) (which has the same cardinality as the reals), I'm going to construct a distinct sequence of rationals that converges to 0.
Think of an element of (0,1) in binary, so as an infinite sequence of 0s and 1s. Fo... | 9 | https://mathoverflow.net/users/1708 | 8200 | 5,618 |
https://mathoverflow.net/questions/8190 | 16 | Is there an notion of elliptic curve over the field with one element? As I learned from a [previous question](https://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids), there are several different versions of what the field with one element and what schemes over it should be (see for example t... | https://mathoverflow.net/users/184 | Elliptic Curves over F_1? | In a strict sense, elliptic curves over the rationals (say) are not defined over $F\_1$ since their reduction modulo $p$ varies with $p$, e.g. they have places of bad reduction. However, CM elliptic curves have some of the properties that one would associate with objects defined over $F\_1$. For example, their L-functi... | 17 | https://mathoverflow.net/users/2290 | 8206 | 5,621 |
https://mathoverflow.net/questions/8202 | 1 | It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves intersect, they do so at a right angle. Does anyone know a good *elementary* proof of this? I.e. no complex analysis, no phys... | https://mathoverflow.net/users/1464 | Systems of conics | As long as you can get an elementary proof of the fact that one of the families consists of ellipses with foci $A=(-c,0)$ and $B=(c,0)$ and the other consists of hyperbolas with the same foci, you can say that for any intersection point $P$ the angle between the lines $PA$ and $PB$ is dissected by the tangent to either... | 4 | https://mathoverflow.net/users/2368 | 8209 | 5,624 |
https://mathoverflow.net/questions/6640 | 6 | What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times\_{\mathbb{F}\_1} \text{Spec } \overline{\mathbb{F}}\_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least one definition of scheme over $\mathbb{F}\_1$.
This really is more a question of opinion.
What do you think this s... | https://mathoverflow.net/users/2024 | What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be? | Sorry I didn't reply before, I somehow didn't read the question till now.
I think your question is a bit misguided. The main problem I see with it is: what is $\text{Spec} \mathbb{Z}[\sqrt{D}]$ over F1? If you think of it as the M\_0 scheme given by $\mathbb{Z}[\sqrt{D}]$ **as a multiplicative monoid**, then it is some... | 3 | https://mathoverflow.net/users/914 | 8215 | 5,629 |
https://mathoverflow.net/questions/8178 | 3 | I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is expressed in terms of schemes, and I believe it is still possible to find some toy example to carry with me without hav... | https://mathoverflow.net/users/1887 | Examples of divisors on an analytical manifold | Hartshorne is the reference where you can find the following example which might be useful.
I what follow everything is with **multiplicity**. Now Alberto pointed out above the case of the divisor over $\mathbb{P}^1$ associated to its "tangent bundle": Two points over the sphere counted with multiplicity (from here tho... | 3 | https://mathoverflow.net/users/1547 | 8222 | 5,634 |
https://mathoverflow.net/questions/8079 | 3 | I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I thought of:
* Vector spaces: Why is it that in linear algebra we always calculate with basis but when we think of a defini... | https://mathoverflow.net/users/nan | Broken Symmetry | Your examples about vector spaces and differential geometry do not make any sense to me.
One does not need coordinates or bases to prove statements in linear algebra and differential geometry.
Personally, I always use coordinate- and basis-free proofs.
For me the reason to avoid coordinates and bases is that we lose ge... | 3 | https://mathoverflow.net/users/402 | 8229 | 5,637 |
https://mathoverflow.net/questions/8203 | 3 | Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every x≠1 from B there is y∈S such that x≤y.
More detailed explanation since apparently my terminology was rather ambiguous:
... | https://mathoverflow.net/users/200 | Semilattices in atomless boolean algebras | The answer to the original question is that no, in fact we can never do this.
**Theorem.** No nontrivial Boolean algebra has a cofinal
subset of B-{1} that is a join-semilattice. Indeed, B cannot have a cofinal subset of B-{1} that is even upward directed.
Proof. Suppose that B is a nontrivial Boolean algebra and ... | 5 | https://mathoverflow.net/users/1946 | 8231 | 5,639 |
https://mathoverflow.net/questions/8232 | 17 | The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a disk packing whose graph is $G$. What happens when disks are replaced by closed balls? By closed balls of higher dimension? I have already asked one question abo... | https://mathoverflow.net/users/1098 | Chromatic number of graphs of tangent closed balls | It's easy to form sets of five mutually-tangent spheres (say, three equal spheres with centers on an equilateral triangle, and two more spheres with their centers on the line perpendicular to the triangle through its centroid). Based on this, I think it should be possible to construct a set of spheres analogous to the ... | 9 | https://mathoverflow.net/users/440 | 8235 | 5,642 |
https://mathoverflow.net/questions/8228 | 3 | Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL\_n(\mathbb{Z})$?
2) What is known about the Dehn function of mapping class groups?
Thanks!!
| https://mathoverflow.net/users/2409 | How do Dehn functions of special linear and mapping class groups behave? | 1) For $SL\_n(\mathbb{Z})$, it depends on the dimension.
a. For $n=2$, the group is virtually free, so the Dehn function is linear.
b. For $n=3$, a theorem of Thurston-Epstein says that it is exponential (this can be found in the book "Word Processing in Groups").
c. For $n>3$, Thurston conjectured that it should... | 8 | https://mathoverflow.net/users/317 | 8237 | 5,643 |
https://mathoverflow.net/questions/8223 | 3 | Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree that -in the sense of connected sum- all these bundles are irreducible?
| https://mathoverflow.net/users/2196 | Reducible 3d torus bundles | Yes - surface bundles over the circle are irreducible (\*) as long as the fiber is not a two-sphere. This follows from the fact that the universal cover of such a surface bundle is homeomorphic to $\mathbb{R}^3$ and Proposition 1.6 of Hatcher's three-manifold [notes](http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.... | 5 | https://mathoverflow.net/users/1650 | 8241 | 5,647 |
https://mathoverflow.net/questions/8225 | 7 | One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (Heyting algebra) of confidence values for the fuzziness. Doing this constructs a fuzzy theory where both the *membership* and ... | https://mathoverflow.net/users/102 | Encoding fuzzy logic with the topos of set-valued sheaves | First I must warn you that there is a difference between fuzzy logics and topos theory. There are some categories of fuzzy sets which are almost toposes, but not quite - they form a quasitopos, which is like a topos, but epi + mono need not imply iso. There is a construction of such a quasitopos in Johnstone's Sketches... | 7 | https://mathoverflow.net/users/1709 | 8249 | 5,652 |
https://mathoverflow.net/questions/8244 | 6 | Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?
Examples of categories of that type abound: **Gr**, **Set**, ...
An specific example of a category where the constraint doesn't hold is given by **Top... | https://mathoverflow.net/users/1593 | What is the name of the following categorical property? | The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category.
However, for maps between sets, "bijective" means "injective and surjective". A common way to interpret "injective" in an arbitrary category is "monic", and a common ... | 35 | https://mathoverflow.net/users/586 | 8250 | 5,653 |
https://mathoverflow.net/questions/8216 | 13 | Most of the group theory that is taught in introductory graduate classes is of the form $$(\mbox{number theory} + \mbox{ group actions} + \mbox{ orbit-stabilizer thm}) + \mbox{group axioms} \Rightarrow \mbox{theorems}$$ So what is the equivalent of "**(number theory + group actions + orbit-stabilizer thm)**" in the mor... | https://mathoverflow.net/users/nan | less elementary group theory | As someone who has spent far too much time thinking about classifying groups of small order, one thing you'd want to do is understand some more results about p-groups. You already probably learned one of the main results, every p-group has a nontrivial center, which is proved by the number theory type arguments you dis... | 17 | https://mathoverflow.net/users/22 | 8283 | 5,675 |
https://mathoverflow.net/questions/8307 | 11 | Basically intuitionistic logic is classical logic minus the law of the excluded middle, i.e. $\neg A\vee A$ is not necessarily valid for all formulas. So I would take this to mean that classical logic allows one to prove more theorems but apparently this view is too naive because yesterday I read the following in a boo... | https://mathoverflow.net/users/nan | intuitionistic interpretation of classical logic | Note that the statements are *classically* equivalent; the intuitionistically derivable statement may be intuitionistically weaker.
For example, the classical ‘law of the excluded middle’ $\lnot A \vee A$ is classically equivalent to $\top$, which is certainly intuitionistically derivable, even though the law of the ... | 6 | https://mathoverflow.net/users/2383 | 8311 | 5,695 |
https://mathoverflow.net/questions/8309 | 2 | Let $k$ be a positive integer greater than $1$ and suppose that $F \in \mathbb{Z}[x\_{1}, \ldots, x\_{k}]$.
Can we always find a natural number $n(k)$ and $f\_{1}, \ldots f\_{n(k)} \in \mathbb{Z}[x]$ such that
$\displaystyle F\Big(\bigoplus\_{j=1}^{k} \mathbb{Z}\Big) = \bigcup\_{j=1}^{n(k)} f\_{j}(\mathbb{Z})$ ?
... | https://mathoverflow.net/users/1593 | Decomposition result for multivariate polynomial | Not really. Take $F(x,y,z,t)=x^2+y^2+z^2+t^2$. Then the image consists of all non-negative integers (Lagrange 4-square theorem). On the other hand, any linear polynomial will give you negatives in the image and every polynomial of degree 2 and higher will give you a zero density set.
| 5 | https://mathoverflow.net/users/1131 | 8312 | 5,696 |
https://mathoverflow.net/questions/8263 | 10 | I have recently become interested in game theory by way of John Conway's on Numbers and Games. Having virtually no prior knowledge of game theory, what is the best place to start?
| https://mathoverflow.net/users/2421 | Just starting with [combinatorial] game theory | *Winning Ways for your Mathematical Plays* (in four volumes) has an enormous amount of stuff about combinatorial games. But most of it you probably won't be interested in for a while. There are a few quickly diverging directions one could study in combinatorial games. Here are some that come to mind immediately, and a ... | 10 | https://mathoverflow.net/users/2046 | 8318 | 5,700 |
https://mathoverflow.net/questions/8236 | 19 | What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded surface. This standard origami has seven axioms which have been proved complete.
The question I have is whether these same... | https://mathoverflow.net/users/2410 | Four Dimensional Origami Axioms | In the $n$-dimensional origami question, you start with a generic set of hyperplanes and their intersections, which can then be some collection of $k$-dimensional planes. An "axiom" is a set of incidence constraints that determines a unique reflection hyperplane, or conceivably a reflection hyperplane that is an isolat... | 14 | https://mathoverflow.net/users/1450 | 8321 | 5,703 |
https://mathoverflow.net/questions/8324 | 43 | All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as **subfields of a larger field**, say $K$. I am happy with the equivalence of the various definitions I've seen in this case. Lang's *Algebra* VIII.3-4 and (thanks to Pete) Zariski & Sa... | https://mathoverflow.net/users/84526 | What does "linearly disjoint" mean for abstract field extensions? | The reasonable meaning following example (1) seems to be that $E \otimes\_k F$ is a field. If so, then it is isomorphic to every compositum. If not, then there exists a compositum within which they are not linearly disjoint.
---
I am not (yet) getting voter support, but I stand my ground! :-)
First, clearly if ... | 26 | https://mathoverflow.net/users/1450 | 8327 | 5,706 |
https://mathoverflow.net/questions/6840 | 10 | For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ring R of characteristic p we can view the Frobenius Fp over R as "the" base change of F from U to R?
I have the follo... | https://mathoverflow.net/users/1841 | Does a universal Frobenius map exist? | I don't really get the categorical picture of what you are asking, but it feels something very similar to the relation between finite fields of characteristic $p$ and the ring (of characteristic 0) of $p$-typical Witt vectors.
You might want to have a look at Borger and Wieland work on [pleythistic algebras](http://w... | 3 | https://mathoverflow.net/users/914 | 8334 | 5,712 |
https://mathoverflow.net/questions/8331 | 4 | What linear algebraic quantities can be calculated precisely for a nonsingular matrix whose entries are only approximately known (say, entries in the matrix are all huge numbers, known up to an accuracy of plus or minus some small number)? Clearly not the determinant or the trace, but probably the signature, and maybe ... | https://mathoverflow.net/users/2051 | Approximately known matrix | SVD is stable, and in some sense incorporates all the stable data you can have, so the answer is: "anything you can see on the SVD". Specifically you can easily see the signature (assuming the matrix is far enough from being singular).
| 7 | https://mathoverflow.net/users/404 | 8336 | 5,714 |
https://mathoverflow.net/questions/8285 | 6 | Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article [link text](http://en.wikipedia.org/wiki/Alain_Badiou) that says:
>
> This effort leads him, in Being and Event, to combine r... | https://mathoverflow.net/users/1977 | Badiou and Mathematics | Badiou's got some mathematical training; reading back and forth between the relevant sections of Goldblatt's *Topoi* and Badiou's account of $\Omega$-sets in *Logics of Worlds*, for example, you can see that the one tracks the other closely. It's not just blind quotation, followed by hand-wavy inference-drawing either:... | 12 | https://mathoverflow.net/users/2451 | 8344 | 5,718 |
https://mathoverflow.net/questions/8182 | 13 | I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum\_{k=0}^{n-1} a\_kx^k\in \mathbb{Z}[x]$ if the condition $|a\_{n-1}|>1+|a\_0|+\cdots+|a\_{n-2}|$ and $a\_0\neq 0$ holds then it is irreduci... | https://mathoverflow.net/users/2384 | Is a polynomial with 1 very large coefficient irreducible? | OK, about your second question. Let's consider the polynomial $x^n+2(x^{n-1}+\ldots+x^2+x)+4$. I claim that this polynomial is almost good for your purposes: if we permute all coefficients except for the leading one, it remains irreducible. Proof: if the constant term becomes 2 after the permutation, use Eisenstein, if... | 4 | https://mathoverflow.net/users/1306 | 8348 | 5,720 |
https://mathoverflow.net/questions/8339 | 8 | Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored over finitely presentable Abelian groups by the following ad hoc construction:
First define $\mathbb{Z}^n \otimes X := \o... | https://mathoverflow.net/users/184 | Tensored Over Abelian Groups? | Given an object $X$ in an additive category $C$ and an abelian group $A$, define the object $A\otimes X$ in $C$ by the rule $Hom\_C(A\otimes X,\:Y) = Hom\_{Ab}(A,Hom\_C(X,Y))$, where $Ab$ denotes the category of abelian groups. If arbitrary direct sums and cokernels (arbitrary colimits, in other words) exist in an addi... | 15 | https://mathoverflow.net/users/2106 | 8349 | 5,721 |
https://mathoverflow.net/questions/7914 | 4 | I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O\_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a ~~globally generated~~ line bundle $L$ such that $L$ is torsion in $Pic(X)$ and $H^1(L) \neq 0$.
Does such surface exist? Ho... | https://mathoverflow.net/users/2083 | Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces | Let us show that a globaly generated torsion line bundle $L$ on a (compact) complex surface is trivial. Ideed, a globally generated line bundle has at least one section, say $s$. Let us take it. If $s$ has no zeros, then $L$ is trivial. But if $s$ vanishes somewhere then any positive power $L^n$ has a section $s^n$ tha... | 4 | https://mathoverflow.net/users/943 | 8355 | 5,726 |
https://mathoverflow.net/questions/8340 | 5 | I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on [nLab](http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories)) of the projective level model structure on symmetric spectra, which has level... | https://mathoverflow.net/users/798 | Are injective Omega-spectra the S-local objects of symmetric spectra for some class S? | You have to realize it has been a long time since we wrote that paper. But I'll give it my best shot.
I think we intentionally chose the injective Omega-spectra because they are "extra fibrant", so to speak. That is, I think S-local spectra don't have to be injective, just Omega-spectra.
The injective Omega-spect... | 6 | https://mathoverflow.net/users/1698 | 8360 | 5,730 |
https://mathoverflow.net/questions/8351 | 14 | Hi Everyone!
I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local class field theory first or uses ideles/cohomology to prove global class field theory. This is not how it... | https://mathoverflow.net/users/1355 | Reference for Learning Global Class Field Theory Using the Original Analytic Proofs? | As far as textbooks your best bet is Janusz's "Algebraic Number Fields."
Also I tried to collect a lot of this stuff in my [senior thesis](http://math.berkeley.edu/~nsnyder/thesismain.pdf). The list of references there should also be very useful. For example, I use Hecke's original approach to abelian L-functions ins... | 12 | https://mathoverflow.net/users/22 | 8364 | 5,732 |
https://mathoverflow.net/questions/8361 | 5 | which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
| https://mathoverflow.net/users/2196 | HNN extensions which are free products | This might help.
**Lemma** If $A$ does not split freely and $C$ is a non-trivial subgroup of $A$ then the HNN extension $G=A\*\_C$ does not split freely.
The proof uses Bass--Serre theory---see Serre's book *Trees* from 1980.
*Proof.* Let $T$ be the Bass--Serre tree of a free splitting of $G$. Because $A$ does no... | 13 | https://mathoverflow.net/users/1463 | 8365 | 5,733 |
https://mathoverflow.net/questions/8345 | 2 | Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$ for all prime $p$ of the form $p=4q+1$ where $q$ is also a prime.
Thanks in advance.
| https://mathoverflow.net/users/808 | Reference of primitive root mod p | Take a look at "A criterion on primitive roots modulo p" by H.Park, J.Park, D.Kim.
There is a collection of various criteria including the above for small primes to appear as primitive roots. I hope it helps, or are you looking for something more general?
| 2 | https://mathoverflow.net/users/2384 | 8366 | 5,734 |
https://mathoverflow.net/questions/8374 | 14 | Using the game of [Battleship](http://en.wikipedia.org/wiki/Battleship_(game)) as an example, is there a general solution for determining the number of arrangements of a given set of 1xN rectangles on a X by Y grid?
**Example:** In Battleship, each player has a 10x10 grid on which they must place each of the followin... | https://mathoverflow.net/users/2455 | Battleship Permutations | 1) Consider a fixed list of boats in the limit of a large grid. This is a model of a battle in the open Pacific, you know; it's not supposed to be Pearl Harbor. Then the inclusion-exclusion formula is much faster than a back-tracking search. Count all of the arrangements of ships, then add or subtract a term for each p... | 16 | https://mathoverflow.net/users/1450 | 8380 | 5,742 |
https://mathoverflow.net/questions/8376 | 7 | Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O\_G$ the structural sheaf of $G$.
Let $D\_e : O\_{G,e} \to k$ a derivation.
I would like to get directly (ie, without any consideration about the cotangent bundle, or some canonical isomorphisms...) a ... | https://mathoverflow.net/users/2330 | A technical question about derivations of sheaves on group schemes | Your derivation at the origin is a map $\mathrm{Spec} k[\epsilon] / \epsilon^2 \rightarrow G$ whose restriction to $\epsilon = 0$ is the inclusion of the origin. This induces a map
$$G \times\_{\operatorname{Spec} k} \operatorname{Spec} k[\epsilon]/\epsilon^2 \rightarrow G \times G \rightarrow G$$
where the first m... | 9 | https://mathoverflow.net/users/32 | 8386 | 5,748 |
https://mathoverflow.net/questions/8367 | 7 | Can anyone describe (or give a reference for) the 2d superspace formulation of N=(2,2) SUSY in Euclidean signature?
I'm reading Hori's excellent introduction to QFT in the book 'Mirror symmetry', and my question is basically Ex. 12.1.1. page 273. What I imagine the answer is is a super version of the usual story of d... | https://mathoverflow.net/users/2454 | N=(2,2) supersymmetry in two-dimensional Euclidean space | You imagine well. Hori is talking about $\mathbb{R}^{2|2}$, which is arguably the simplest *super Riemann surface*.
There is lots on this subject, mostly in the Physics literature, which I'm hesitant to recommend. In the Mathematics literature, you might wish to read Deligne and Freed's [Supersolutions](http://arxiv.... | 5 | https://mathoverflow.net/users/394 | 8387 | 5,749 |
https://mathoverflow.net/questions/7058 | 8 | In *Proofs and Types*, Girard discusses coherent (or coherence) spaces, which is defined as a set family which is closed downward ($a\in A,b\subseteq a\Rightarrow b\in A$), and binary complete (If $M\subseteq A$ and $\forall a\_1,a\_2\in M (a\_1\cup a\_2\in A)$, then $\cup M\in A$)
It was informally related to topolo... | https://mathoverflow.net/users/2143 | Coherent spaces | An analogue of your coherence spaces are used extensively in set theory, particularly with the method of forcing, the set-theoretic technique often used to prove statements independent of ZFC. But there is a variation, in that the coherence clause is weakened to cover only some M, such as M of a certain size.
For ex... | 2 | https://mathoverflow.net/users/1946 | 8389 | 5,750 |
https://mathoverflow.net/questions/7350 | 21 | <http://en.wikipedia.org/wiki/Axiom_of_dependent_choice>
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full development of the theory of von Neumann algebras on a separable Hilbert space?
| https://mathoverflow.net/users/2206 | Is Dependent Choice all we really need? | Let me adopt an extreme interpretation of your question, in order to prove an affirmative answer.
Yes, in the arena of the countable, DC suffices.
To see why, let me first explain what I mean. If one wants to consider only countable sets, then the natural set-theoretic context is HC, the class of hereditarily cou... | 23 | https://mathoverflow.net/users/1946 | 8392 | 5,752 |
https://mathoverflow.net/questions/8396 | 31 | Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself.
Say I have a nasty ODE, nonlinear, maybe extremely singular. It showed up naturally mathematically (I'm actually thinking of Painleve VI, which comes from isomonodromy represe... | https://mathoverflow.net/users/622 | Does every ODE comes from something in physics? | It is possible to solve a large class of ODEs by means of analog computers. Each of the pieces of the differential equation corresponds to an electronic component and if you wire them up the right way you get a circuit described by the ODE. [Wikipedia](https://en.wikipedia.org/wiki/Analog_computer) has lots of informat... | 18 | https://mathoverflow.net/users/1233 | 8398 | 5,756 |
https://mathoverflow.net/questions/8407 | 6 | Is there a generalized method to find the projective closure of an affine curve? For example, I read that the projective closure of $y^2 = x^3−x+1$ in $\mathbb{P}^2$ is $y^2z = x^3−xz^2+z^3$.
If I want to find the the closure of another affine curve, what method should I employ? I can't seem to find an adequate descr... | https://mathoverflow.net/users/2473 | Projective closure of affine curve | When the curve is a plane curve of degree $d$, the formula is simple (and in fact, this works for any hypersurface) you take $f(x,y)=0$ and replace it by $z^d f(x/z,y/z)=0$. This will be homogeneous of degree $d$, and when $z\neq 0$, you recover your curve.
Now, if you have a more general affine variety, given by $\l... | 26 | https://mathoverflow.net/users/622 | 8410 | 5,763 |
https://mathoverflow.net/questions/8415 | 24 | The number of conjugacy classes in $S\_n$ is given by the number of partitions of $n$. Do other families of finite groups have a highly combinatorial structure to their number of conjugacy classes? For example, how much is known about conjugacy classes in $A\_n$?
| https://mathoverflow.net/users/960 | Combinatorial Techniques for Counting Conjugacy Classes | Since $A\_n$ has index two in $S\_n$, every conjugacy class in $S\_n$ either is a conjugacy class in $A\_n$, or it splits into two conjugacy classes, or it misses $A\_n$ if it is an odd permutation. Which happens when is a nice undergraduate exercise in group theory. (And you are a nice undergraduate. :-) )
The pair ... | 31 | https://mathoverflow.net/users/1450 | 8418 | 5,768 |
https://mathoverflow.net/questions/8395 | 4 | So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup:
I've been looking at the simplest case that didn't look completely trivial: two vertices with one directed edge. Now, my understanding is that then we have two vector spaces $V$... | https://mathoverflow.net/users/622 | Near Trivial Quiver Varieties | First, one important point: people who study quiver varieties seem not to usually take stack quotients, but rather GIT quotients (though there ARE very good reasons to do this if you like geometric representation theory) which leads them to come up different dimension formulae from you. The reason is that you are think... | 7 | https://mathoverflow.net/users/66 | 8420 | 5,770 |
https://mathoverflow.net/questions/7016 | 27 | Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it true that $F$ is a round disk?
**Comments:**
* It is easy to see that the **round disk** has this property.
* One ca... | https://mathoverflow.net/users/1441 | When shorter means smaller? | Here's a few thoughts on the question:
First, although this is probably obvious to everyone who's posted already, the region F must be bounded (if it is not $\mathbb{R}^2$). If not, then since it's convex, it must contain an infinite ray, and F must be contained in a half-space since it is convex. Take the projection... | 3 | https://mathoverflow.net/users/1345 | 8424 | 5,773 |
https://mathoverflow.net/questions/8414 | 2 | The question is pretty self-explanatory; we are dealing with the standard symplectic structure on ℝ4.
Some background: I'm reading the thesis "Lagrangian Unknottedness of Tori in Certain Symplectic 4-manifolds" by Alexander Ivrii, which proves that all embedded Lagrangian tori in ℝ4 are smoothly isotopic (and, in fac... | https://mathoverflow.net/users/2467 | Why (and whether) is any smooth embedded torus in R^4 isotopic to an embedded Lagrangian torus? | Whoever told you that any embedded torus in R4 is isotopic to a Lagrangian torus was sorely mistaken. Luttinger (JDG 1995) observed the following: The manifolds obtained by doing certain Dehn-type surgeries on a Lagrangian torus in R4 admit symplectic structures. But the result X of the surgery is then always minimal a... | 13 | https://mathoverflow.net/users/424 | 8425 | 5,774 |
https://mathoverflow.net/questions/8428 | 5 | We can embed $S^2\times I$ into $\mathbb{R}^3$ by taking a compact 3-ball and removing an open 3-ball from its interior. Taking the boundary gives an embedding $i: S^2\sqcup S^2\hookrightarrow\mathbb{R}^3$, as "a sphere contained inside another sphere". Now it's intuitively clear that this embedding is not ambient-isot... | https://mathoverflow.net/users/1182 | Freeing a sphere from within a sphere | **For your first question**, look at complements (a "fundamental" technique in analyzing ambient isotopies/ambient homeomorphisms; e.g. the "[knot group](http://en.wikipedia.org/wiki/Knot_group)"):
Let the union of the initial two spheres be $S$, and the union of the final two spheres be $T$. An isotopy on $\mathbb{R... | 9 | https://mathoverflow.net/users/84526 | 8430 | 5,776 |
https://mathoverflow.net/questions/8426 | 4 | Using Vandermonde's identity we know:
$\sum\_{i=0}^k \binom{k}{i}\binom{n-k}{n/2-i} = \binom{n}{n/2}$.
I'm interested in how close the alternating sum is to 0 when k << n. I.e.,
$\sum\_{i=0}^k (-1)^i\binom{k}{i}\binom{n-k}{n/2-i}$.
| https://mathoverflow.net/users/2476 | alternating sums of terms of the Vandermonde identity | So, you are interested in $f(n,k)=\sum\_{i=0}^k (-1)^i\binom{k}{i}\binom{2n-k}{n-i}$.
Simple manipulations show $f(n,k)=\frac{k!(2n-k)!}{(n!)^2}\left[\sum\_{i=0}^n (-1)^i \binom{n}{i}\binom{n}{k-i}\right]$
Now the second factor counts the coefficient of $x^k$ in $(1-x^2)^n$ and therefore if $k$ is odd $f=0$ otherwise $... | 5 | https://mathoverflow.net/users/2384 | 8431 | 5,777 |
https://mathoverflow.net/questions/8445 | 25 | **EDIT (Harry):** Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ahead and post with regard to that topic, rather than graph theory, which the questioner can ask again in another topic... | https://mathoverflow.net/users/2482 | Learning Topology | 1. A self study course I can recommend for topology is **Topology** by **JR Munkres** followed by **Algebraic Topology** by **A Hatcher** (freely and legally available online, courtesy of the author!). But that is if you want to be able to really do the math in all its glorious detail. **Basic Topology** by **MA Armstr... | 20 | https://mathoverflow.net/users/262 | 8452 | 5,789 |
https://mathoverflow.net/questions/8451 | 12 | Let $A \to B$ be a ring extension.
What is the definition of $B/A$ étale ?
When $A$ is a field, do we get a nice characterization ?
| https://mathoverflow.net/users/2330 | Definition of étale for rings | You say that a ring homomorphism $\phi: A \to B$ is **étale** (resp. **smooth**, **unramified**), or that $B$ is étale (resp. smooth, unramified) over $A$ is the following two conditions are satisfied:
* $A \to B$ is **formally étale** (resp. **formally smooth**, **formally unramified**): for every square-zero extens... | 18 | https://mathoverflow.net/users/1797 | 8455 | 5,792 |
https://mathoverflow.net/questions/8460 | 22 | In Weibel's *An Introduction to Homological Algebra*, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^\*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. The differential here has degree -1.
I have been told that the Chevalley-Eilenberg complex for $g$ is
$C^\*(g) = ... | https://mathoverflow.net/users/1676 | Chevalley Eilenberg complex definitions? | The first complex, from Weibel, is a projective resolution of the trivial $\mathfrak g$-module $k$ as a $\mathcal U(\mathfrak g)$-module; I am sure Weibel says so!
Your second complex is obtained from the first by applying the functor $\hom\_{\mathcal U(\mathfrak g)}(\mathord-,k)$, where $k$ is the trivial $\mathfrak... | 21 | https://mathoverflow.net/users/1409 | 8461 | 5,797 |
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