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https://mathoverflow.net/questions/21088 | 4 | I am given a triple of positive integers $a,b,c$ such that $a \geq 1$ and $b,c \geq 2$.
I would like to find an upper bound for $a+b+c$ in terms of $n = ab+bc+ac$. Clearly $a+b+c < ab+bc+ac = n$.
Is there any sharper upper bound that could be obtained (perhaps asimptotically)?
| https://mathoverflow.net/users/1737 | Upper bound for a+b+c in terms of ab+bc+ac | This is (mostly) a pretty routine optimization problem. The methods of (for example) a standard calculus class are enough to tell you that $a+b+c$ will be largest when two of $a,b,c$ are as small as possible and the third is whatever it has to be. So if you don't care whether the variables are integers, take $a=1,b=2,c... | 12 | https://mathoverflow.net/users/2559 | 21098 | 13,982 |
https://mathoverflow.net/questions/21090 | 6 | Assume $M$ is a compact smooth manifold (without boundary). What can we say about the spectrum of the $\mathbb{R}$-algebra $A=C^{\infty}(M)$? The elements of $M$ give rise to rational points of $A$, are there other ones? Does the smooth structure of $M$ endow $A$ with additional structure such that $M$ can be completel... | https://mathoverflow.net/users/2841 | smooth Gelfand-duality | Perhaps I should post this as an answer (even if I don't really know that theory): in
Juan A. Navarro González & Juan B. Sancho de Salas, C∞-Differentiable Spaces, LNM 1824 [Google Books Preview](http://books.google.it/books?id=79bCwXbVnHAC&printsec=frontcover&dq=C-infinity+differentiable+spaces&source=bl&ots=xjtnnn... | 4 | https://mathoverflow.net/users/4721 | 21099 | 13,983 |
https://mathoverflow.net/questions/21087 | 8 | Call a subset of $\mathbb{N}$ *primitive-recursively enumerable (p-r.e.)* if it is empty or an image of a primitive recursive function. I feel like a lot must be known about the poset of such sets ordered by inclusion, but I am unable to dig up references. Concretely, I would like to know whether there exists a p-r.e. ... | https://mathoverflow.net/users/1176 | Is there a "primitive-recursively enumerable" set whose complement is not such? | There is a stronger result: *Every r.e. set is primitive r.e. in your sense.*
Short proof: Kleene's Normal Form Theorem.
Longer proof: Let *S* be an r.e. set, assumed WLOG nonempty; fix *a* ∈ *S*, and fix an algorithm *e* where *S* is precisely the range of the function computed by *e*.
Consider the following alg... | 8 | https://mathoverflow.net/users/4133 | 21101 | 13,984 |
https://mathoverflow.net/questions/9785 | 13 | Let $R$ be a local normal domain, and let $P \in Spec (R)$. It is well known that $Cl(R) \to Cl(R\_P)$ is surjective. However, I do not know any example where $Cl(R)$ is torsion-free, but $Cl(R\_P)$ is not (we consider $(0)$ to be torsion-free). So:
If the class group of $R$ is torsion-free, must all the class groups... | https://mathoverflow.net/users/2083 | Does torsion-freeness of class group localize? | Perhaps something like the following works (I have not checked all the details):
Let $C$ be a smooth plane conic and let $Y$ be the projective cone over $C$. Then $Cl(Y) = \mathbb{Z}$ but the class group of the local ring of the vertex of $Y$ is $\mathbb{Z}/2$. Let $X$ be affine cone over $Y$. Then the class group of... | 3 | https://mathoverflow.net/users/519 | 21106 | 13,989 |
https://mathoverflow.net/questions/20956 | 26 | I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The question is waay down there.
Let $f$ be a cuspidal modular eigenform of level $\Gamma\_0(N)\subseteq SL\_2(\mathbf{Z}... | https://mathoverflow.net/users/1384 | Are there any Hecke operators acting on an elliptic curve with additive reduction that I don't know about? | Just to expand on a comment I made above: I'm not exactly sure what operators generate the Hecke algebra of $\Gamma\_0(p^2)$, but the Hecke algebras of the principal congruence subgroups $\Gamma(p^r)$ are easier to handle.
Let's write $K\_n$ for the principal congruence subgroup of level $p^n$ in $G = {\rm GL}\_2(\m... | 12 | https://mathoverflow.net/users/2481 | 21108 | 13,990 |
https://mathoverflow.net/questions/21110 | 35 | I only know of one proof of Hilbert 90, which is very smart if not magical. See for example <http://hilbertthm90.wordpress.com/2008/12/11/hilberts-theorem-90the-math/>
Does anyone know of a more intuitive proof or know a good way to view the proof?
I have accepted the answer by Emerton, great thanks as well to Davi... | https://mathoverflow.net/users/2701 | Is there a natural way to view the proof of Hilbert 90? | Here is a proof of Hilbert's Theorem 90 in the case of cyclic extensions which
I think is fairly conceptual. The key point (which is also at the heart of Grothendieck's
very general version in terms of flat descent) is that if we want to verify that
a linear transformation has a certain eigenvalue (in our particular c... | 38 | https://mathoverflow.net/users/2874 | 21117 | 13,995 |
https://mathoverflow.net/questions/21114 | 9 | This is a reference request question. I would like to know more on the structure of low dimensional nilpotent lie algebras. I heard that up to dimension 6 there are only finitely many isomorphism classes, and every such algebra admits a gradation with only positive degrees (see <http://en.wikipedia.org/wiki/Graded_Lie_... | https://mathoverflow.net/users/1049 | Low dimensional nilpotent Lie algebras | Classification of nilpotent Lie algebras in characteristic 0 is an old problem,
with a lot of literature. For the dimensions up to 6 there is a finite list.
Among the many relevant papers on MathSciNet, I'll list just a few:
MR2372566 (2009a:17027) 17B50 (17B20 17B30)
Strade, H. (D-HAMBMI)
Lie algebras of small dimen... | 17 | https://mathoverflow.net/users/4231 | 21118 | 13,996 |
https://mathoverflow.net/questions/17189 | 29 | The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise content.
If $a$ and $b$ are nonnegative real numbers such that $a+b=1$, show that $a^{2b} + b^{2a}\le 1$.
| https://mathoverflow.net/users/3818 | Is there a good reason why $a^{2b} + b^{2a} \le 1$ when $a+b=1$? | Fixed now. I spent some time looking for some clever trick but the most unimaginative way turned out to be the best. So, as I said before, the straightforward Taylor series expansion does it in no time.
Assume that $a>b$. Put $t=a-b=1-2b$.
Step 1:
$$
\begin{aligned}
a^{2b}&=(1-b)^{1-t}=1-b(1-t)-t(1-t)\left[\frac{... | 37 | https://mathoverflow.net/users/1131 | 21126 | 14,000 |
https://mathoverflow.net/questions/21081 | 7 | Let $X$ be a regular scheme which is flat over $\mathbf{Z}$. The arithmetic Grothendieck group $\hat{K}(X)$ is defined to be the quotient of $\hat{G}(X)$ by $\hat{G}^\prime(X)$. This is actually quite a length definition which I added below for the sake of completeness.
In the classical case, for $X$ any noetherian s... | https://mathoverflow.net/users/4333 | Is there a category-theoretic definition of the arithmetic Grothendieck group | The classical group $K\_0$ can also be thought of as consisting of equivalence classes of chain complexes of vector bundles, such that the *exact* sequences represent the zero of $K\_0$ --- and furthermore every complex is equivalent to a two-term complex, a.k.a. a bundle morphism; the graded tensor product of complexe... | 2 | https://mathoverflow.net/users/1631 | 21131 | 14,003 |
https://mathoverflow.net/questions/21142 | 2 | To assuage my conscience over an unsourced statement in a paper I'm writing:
I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence.
| https://mathoverflow.net/users/5316 | Maximal ideal of codimension >1 | If the codimension is finite, then there is no such thing. If the codimension can be infinite, then yes, because there are infinite dimensional complex division algebras which are simple, like $\mathbb C(t)$.
| 6 | https://mathoverflow.net/users/1409 | 21144 | 14,011 |
https://mathoverflow.net/questions/21109 | 9 | Is there a non-trivial class $S$ of smooth Deligne-Mumford stacks over a base $B$ with the property that if $\mathcal{X}, \mathcal{Y} \in S$ have isomorphic coarse moduli spaces (assumed to exist) then $\mathcal{X} \cong \mathcal{Y}$? If $B = Spec(k)$, $k$ a field (of characteristic $0$ if necessary), can one take $S$ ... | https://mathoverflow.net/users/519 | Stacks determined by their coarse moduli spaces | Yes, the class of all smooth, separated DM stacks over a field of characteristic $0$, with trivial inertia in codimension at most $1$, over a field of characteristic $0$, has the propery you want. The point is that every moduli space of such a stack has quotient singularities; and every variety with quotient singularit... | 18 | https://mathoverflow.net/users/4790 | 21146 | 14,013 |
https://mathoverflow.net/questions/21134 | 5 | Suppose $K$ is a local field and $L$ a finite cyclic extension of $K$. By Hilbert 90, we know that if an element $a$ in $ L$ such that $N\_{L / K}(a) =1 $ then $ a = b / \sigma(b) $ for some $b$ in $L$ and $ \sigma$ a generator of the Galois group.
My question: Suppose $N\_{L / K}(a) \simeq 1 $, is there some $b$ suc... | https://mathoverflow.net/users/2701 | Is there any approximated version of Hilbert 90? | Yes such approximated versions of Hilbert 90 do exist. But you need some technical conditions.
For instance assume that $L/K$ is unramified of degree $d$ and that
$a\in {\mathfrak o}\_{L}^{\times}$.
Then you condition writes
$N\_{L/K}(a)\equiv 1$ modulo $\mathfrak{p}\_{K}^{n}$,
for some $n>0$ (I assume that ... | 8 | https://mathoverflow.net/users/4767 | 21156 | 14,020 |
https://mathoverflow.net/questions/21161 | 15 | Artin's presentation of **braid group on three strands** is:
$$ B\_3 = \langle l,r : lrl = rlr \rangle $$
where you should think of "$l$" as the positive crossing between the left and middle strands and "$r$" as the positive crossing of the right and middle strands:
```
| | | | | |
\ / ... | https://mathoverflow.net/users/78 | Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid? | No.
Let $P\_n$ be the pure braid group on $n$ strands. Forgetting the first strand gives a projection $P\_3\to P\_2$. The kernel is the free group $F\_2=\pi\_1(\mathbb{C}\setminus
\{2,3\},1)$ given by moving the first strand around the other two while they are held in place. Since $P\_2$ is just $\mathbb{Z}$, we have... | 17 | https://mathoverflow.net/users/250 | 21166 | 14,027 |
https://mathoverflow.net/questions/21152 | 33 | A **magma** is a set $M$ equipped with a binary operation $\* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional axioms that "symmetries of things" should satisfy. This is made precise in the sense that for any object $A$ in a categ... | https://mathoverflow.net/users/290 | Do non-associative objects have a natural notion of representation? | Since magmas in general don't have much structure, we can't reasonably expect a representation to preserve much structure. We can therefore define a left representation of a magma $M$ to be a set $V$ equipped with a map $M \times V \to V$. We do the analogous thing for general nonassociative algebras. Serge Lang liked ... | 13 | https://mathoverflow.net/users/121 | 21172 | 14,029 |
https://mathoverflow.net/questions/21171 | 43 |
>
> **Possible Duplicate:**
>
> [Cohomology and fundamental classes](https://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes)
>
>
>
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi\_\*[N]\in H\_\*(M)$ where $[N]$ is the fundament... | https://mathoverflow.net/users/5323 | When is a Homology Class Represented by a Submanifold? | Here are a few simple answers to the question you asked:
1. Every class in $H\_{n-1}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to S^1$ representing the Poincare dual in $H^1(M;Z)=[M,S^1]$ and take the preimage of a point. In dimensions>2 it can be taken connected.
2. Similarly... | 31 | https://mathoverflow.net/users/3874 | 21198 | 14,044 |
https://mathoverflow.net/questions/21200 | 12 | Brauer's permutation lemma states that any two permutation matrices are conjugate in $\mathrm{GL}(n,\mathbb{C})$ if and only if they are conjugate in the symmetric group, i.e., they have the same cycle type (we can replace $\mathbb{C}$ by any field of characteristic zero).
Another way of stating it is that any two pe... | https://mathoverflow.net/users/3040 | Brauer's permutation lemma — extending to some other finite groups? | There are many group actions on sets which are linearly equivalent but not equivalent as actions. In fact, every group other than the cyclic group has one. This follows from some easy linear algebra:
* the number of irreducible reps over $\mathbb{Q}$ is the number of conjugacy classes of cyclic subgroups of $G$, (**E... | 11 | https://mathoverflow.net/users/66 | 21204 | 14,045 |
https://mathoverflow.net/questions/21199 | 5 | A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). So far I have been unable to find a proof of this theorem anywhere. The only references I have seen are to Isaacs' book ... | https://mathoverflow.net/users/4614 | Reference request: A theorem by S. Garrison | A new proof was published in:
Isaacs, I. M.; Knutson, Greg. "Irreducible character degrees and normal subgroups."
J. Algebra 199 (1998), no. 1, 302–326.
[MR1489366](http://www.ams.org/mathscinet-getitem?mr=1489366)
[DOI:10.1006/jabr.1997.7191](http://dx.doi.org/10.1006/jabr.1997.7191)
This was extended to cd(G)=5 i... | 7 | https://mathoverflow.net/users/3710 | 21215 | 14,051 |
https://mathoverflow.net/questions/21207 | 9 | Faber's perfect pairing conjecture states that the tautological ring $R^\*$ of the moduli space $\mathcal{M}\_g$ of curves of genus $g$ behaves as if it were the rational cohomology of a closed, oriented manifold of dimension $g-2$. Specifically, $R^{g-2}$ is rank one, and multiplication into this degree gives a perfec... | https://mathoverflow.net/users/4649 | Applications of Faber's conjecture | 1. Numerical evidence, from computing the cases $g=2,3,\dots$, eventually $g\le 15$, and seeing the symmetry in the numbers $\dim R\_g^n$. I recall Carel saying he made the conjecture when $g$ was still pretty low, maybe 6. For any $g$, there is an algorithm computing $\dim R^n\_g$ in finite time, that Faber came up wi... | 6 | https://mathoverflow.net/users/1784 | 21222 | 14,057 |
https://mathoverflow.net/questions/21168 | 17 | This continues my question about [smooth Gelfand-duality](https://mathoverflow.net/questions/21090/smooth-gelfand-duality). In the book
>
> Juan A. Navarro González & Juan B. Sancho de Salas, C∞-Differentiable Spaces, LNM 1824
>
>
>
it is shown that $M \mapsto C^\infty(M)$ is a fully faithfull contravariant fu... | https://mathoverflow.net/users/2841 | How to classify the algebras C^∞(M)? | There is a strongly geometric characterization of those algebras which arise as $C^\infty(M)$ for $M$ compact and orientable, recently proved by Connes, see
[here](http://arxiv.org/abs/0810.2088). This has come up on MO before, e.g. in Joel Fine's answer to this question:
[Algebraic description of compact smooth man... | 10 | https://mathoverflow.net/users/2356 | 21230 | 14,063 |
https://mathoverflow.net/questions/20980 | 5 | It is well known that for a closed hyperbolic 3-manifold $M$ the rank of $\pi\_1(M)$ is bounded above by some universal constant $K$ times the volume of $M$. Using similar methods, i.e. the thick-thin decomposition of $M$, one can also show that the Heegaard genus of $M$ is bounded above by a universal constant times t... | https://mathoverflow.net/users/2788 | Heegaard genus in hyperbolic 3-manifolds | Agol says: Take a 3-manifold group that maps onto a free group, and take induced covers (the rank and volume will both grow linearly). See [this paper of Lackenby](http://ams.org/mathscinet-getitem?mr=2218779).
| 2 | https://mathoverflow.net/users/66 | 21240 | 14,069 |
https://mathoverflow.net/questions/21238 | 1 | Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a\_n = \sum{{b\_i}{a\_{n-i}}} + f(n)$ )
| https://mathoverflow.net/users/2973 | Inhomogenous recurrence relations | One standard method is generating functions. Set $A(t)=\sum\_{n=0}^\infty a\_n t^n$
and $B(t)=\sum\_{i=1}^\infty b\_i t^i$. Then
$$A(t)=a\_0+B(t)A(t)+\sum\_n f(n)t^n$$
so that
$$A(t)=(1-B(t))^{-1}\left(a\_0+\sum\_n f(n)t^n\right).$$
For an excellent text on generating functions, see Herb Wilf's
*generatingfunctionology... | 6 | https://mathoverflow.net/users/4213 | 21241 | 14,070 |
https://mathoverflow.net/questions/21205 | 6 | Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$?
Edit: In fact I'm interested in nef line bundles D, not just divisors.
| https://mathoverflow.net/users/nan | Nef divisors with few global sections | Here's an interesting example from complex algebraic surfaces: the so called Godeaux surface. This is a surface, $S$ on which the canonical bundle is $ample$ and yet $h^{0}(S,K\_{S})=0$.
To construct such an $S$ we start with a quintic, $S'$ in $\mathbb{P}^{3}$ defined by the Fermat form:
$X\_{0}^{5}+X\_{1}^{5}+X\_... | 11 | https://mathoverflow.net/users/5124 | 21254 | 14,078 |
https://mathoverflow.net/questions/21243 | -1 | Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in [utility theory](http://en.wikipedia.org/wiki/Useful) from economics (but originally from the Bernoullis and improved by von Neumann, so very much 'mathematical'... | https://mathoverflow.net/users/3993 | Formal definition of 'useful' ? | It seems to me that you have two questions here.
First, you inquire about a formal account of "usefulness". I believe that this is already provided by the formal mathematical accounts of utility in utility theory. The concept of utility in that theory is extremely flexible, not limited to economics or any other spec... | 3 | https://mathoverflow.net/users/1946 | 21256 | 14,080 |
https://mathoverflow.net/questions/21258 | 4 | Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi \rightarrow V $ is a vector bundle of rank $2n$ admitting the symplectic structure $d\alpha$. Also consider the Reeb ve... | https://mathoverflow.net/users/5259 | Looking for almost complex structure on a contact manifold invariant under flow of Reeb vector field !? | In general there is no invariant complex structure.
Let $\gamma$ be a closed orbit of the Reeb field. Consider a linearization $A$ of the Poincare return map along $\gamma$. $A$ is not, in general, a realification of a complex operator (with respect to any arbitrary complex structure). For example, as far as I rememb... | 8 | https://mathoverflow.net/users/2823 | 21268 | 14,086 |
https://mathoverflow.net/questions/21217 | 0 | The following question naturally originates from [this question](https://mathoverflow.net/questions/21168/how-to-classify-the-algebras-cm)
and [this one](https://mathoverflow.net/questions/21090/smooth-gelfand-duality).
While the usual $C^{0}$ Gelfand duality involves a topology on the function algebras considered (i... | https://mathoverflow.net/users/4721 | Why "smooth Gelfand duality" does not involve a topology on the algebras? | We don't need a topology because the proofs show that the space can be recovered without a topology. I think it's just that simple. Perhaps this is not satisfactory for you?
Continuing George's comment, there is an important result in the theory of $C^\\*$-algebras: the norm is unique. The reason is that every $C^\\*... | 3 | https://mathoverflow.net/users/2841 | 21271 | 14,088 |
https://mathoverflow.net/questions/21267 | 39 | A number field $K$ is said to be [monogenic](http://en.wikipedia.org/wiki/Monogenic_field) when $\mathcal{O}\_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}\_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus's *Number Fields*, I'm familiar with the proof that the cyclotomic fie... | https://mathoverflow.net/users/1916 | Which number fields are monogenic? and related questions | Zev, when $[K:{\mathbf Q}] > 2$, finding all $\alpha$ which are ring generators for ${\mathcal O}\_K$ is a hard problem in general: there are only finitely many choices modulo the obvious condition that if
$\alpha$ works then so does $a + \alpha$ for any integer $a$. In other words, up to adding an integer there are o... | 45 | https://mathoverflow.net/users/3272 | 21284 | 14,097 |
https://mathoverflow.net/questions/21247 | 4 | I am trying to use the Chebotarev Density Theorem to say something about the Galois groups of a class of polynomials. To be more precise, by factoring a polynomial mod some prime p, I want to show that there is an element in the Galois group of the polynomial with a certain cycle structure. Unfortunately my knowledge o... | https://mathoverflow.net/users/4078 | Ramified primes in the Chebotarev Density Theorem | Adam, the requirement that $p$ be unramified in the number field is to explain the existence of an element (really, conjugacy class) in the Galois group with a certain cycle structure on the roots of a generator for the number field. The way this element of the Galois group is constructed requires algebraic number theo... | 12 | https://mathoverflow.net/users/3272 | 21289 | 14,099 |
https://mathoverflow.net/questions/21290 | 17 | Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).
| https://mathoverflow.net/users/5337 | What's an example of a transcendental power series? | If $k$ has characteristic zero, then $\displaystyle e^t = \sum\_{n \ge 0} \frac{t^n}{n!}$ is certainly transcendental over $k[t]$; the proof is essentially by repeated formal differentiation of any purported algebraic relation satisfied by $e^t$.
**Edit:** Let me fill in a few details. Given a polynomial $P$ in $e^t... | 26 | https://mathoverflow.net/users/290 | 21291 | 14,100 |
https://mathoverflow.net/questions/21286 | 11 | In his book *Commutative Ring Theory*, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional.
What is an example of a local ring which does not admit such a presentation?
| https://mathoverflow.net/users/1594 | A local ring not a quotient of a regular local ring | A source available online is this [paper](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-70/issue-none/Examples-of-bad-Noetherian-local-rings/nmj/1118785530.full) "Examples of bad Noetherian rings" by Marinari (example 2.1).
The reason many of these types of construction work is because of the ... | 15 | https://mathoverflow.net/users/2083 | 21304 | 14,105 |
https://mathoverflow.net/questions/21121 | 14 | The homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more accessible Luminy notes of Bisson-Joyal). Has anyone used this to construct a spectral sequence converging under some assum... | https://mathoverflow.net/users/4991 | Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop spaces (spectra). | This might not quite be what you're looking for, Dev, but you should check out Paul Goerss and Mike Hopkins' "Multiplicative ring spectra project," on Paul's [webpage](http://www.math.northwestern.edu/~pgoerss/). They construct such a spectral sequence using Andre-Quillen cohomology in "Moduli spaces of commutative rin... | 9 | https://mathoverflow.net/users/4649 | 21305 | 14,106 |
https://mathoverflow.net/questions/21310 | 2 | Given a planar graph $G=(V,E)$ with vertices $V$ and edges $E$, call $\bar G = (V,\bar E)$ a non-planar extension of $G$ if $\bar G$ is non-planar and $E \subset \bar E$.
I'm interested in minimal non-planar extensions in the sense that if $\bar G$ is a non-planar extension of $G$, there is no non-planar extension of... | https://mathoverflow.net/users/5312 | Minimal Non-planar Extensions of a Graph | (1) and (2) Take the union of a $K\_5$ missing an edge and a $K\_{3,3}$ missing an edge. This graph has 2 minimal non-planar extensions, which are obviously not isomorphic.
| 6 | https://mathoverflow.net/users/3684 | 21311 | 14,109 |
https://mathoverflow.net/questions/21295 | 37 | Let $(\mathcal{M},g)$ be a $C^{\infty}$-Riemannian manifold. A basic fact is that $g$ endows the manifold $\mathcal{M}$ with a metric space structure, that is, we can define a distance function $d:\mathcal{M}\times\mathcal{M}\longrightarrow\mathbb{R}$ (the distance between two points will be the infimum of the lengths ... | https://mathoverflow.net/users/5069 | Smoothness of distance function in Riemannian Manifolds | As others mentioned, you have to remove the diagonal of $M\times M$ or square the distance function. Then, for a complete $M$, the answer is the following.
The distance function is differentiable at $(p,q)\in M\times M$ if and only if there is a *unique* length-minimizing geodesic from $p$ to $q$. Furthermore, the di... | 64 | https://mathoverflow.net/users/4354 | 21316 | 14,110 |
https://mathoverflow.net/questions/21315 | 9 | On a Kahler manifold, the different Laplacians are compatible: $\Delta\_d=2\Delta\_{\bar{\partial}}=2\Delta\_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
| https://mathoverflow.net/users/nan | Non-Kahler manifolds where the different Laplacians are compatible | Hermitian manifolds $M$ where $$\Delta\_d f=2\Delta\_{\bar{\partial}} f=2\Delta\_{\partial} f$$ holds for every smooth function $f$ on $M$ are called *balanced*.
For more information, you can search for "balanced hermitian manifolds", [Here](http://cdsweb.cern.ch/record/422889/files/), for instance, is a paper that r... | 12 | https://mathoverflow.net/users/2384 | 21317 | 14,111 |
https://mathoverflow.net/questions/17641 | 8 | Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-braidings $V \otimes W \rightarrow W \otimes V$ for every object W in C satisfying some naturality conditions. Hence there... | https://mathoverflow.net/users/22 | Is there a source for a diagrammatic description of the induction functor C->Z(C)? | In the meanwhile a reference has appeared: Theorem 2.3 in <http://arxiv.org/abs/1004.1533>
| 2 | https://mathoverflow.net/users/1035 | 21320 | 14,113 |
https://mathoverflow.net/questions/21314 | 15 | Motivated by [this question](https://mathoverflow.net/questions/21290/whats-an-example-of-a-transendental-power-series), I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an example of a power series th... | https://mathoverflow.net/users/1306 | An example of a series that is not differentially algebraic? | You'd be better off in characteristic zero, for $f^{(p)}(t)=0$ in characteristic $p$. Then the sea of zeroes example $\sum\_n t^{n^n}$ will do the trick. For large enough $n$, there will be a cluster of non-zeroes in degrees $kn^n-m$ for small (and bounded) $k$ and $m$, "reachable" only by products of $(t^{n^n})^{(s)}$... | 7 | https://mathoverflow.net/users/5301 | 21321 | 14,114 |
https://mathoverflow.net/questions/11183 | 6 | I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of the edges on the hypercube could be different for different dimensions. The hypercube basically is a hyper-rectangle.
Now... | https://mathoverflow.net/users/2013 | How can I embed an N-points metric space to a hypercube with low distortion? | Y. Bartal has studied a related problem of embedding metric spaces to *hierarchically separated trees*. With $1 < \mu$ being a fixed real number, a $\mu$-HST is equivalent to the set of corners of a rectangle whose edges are of length $c, c\mu^{-1}, c\mu^{-2}, \dots, c\mu^{1-D}$ with the $l\_\infty$-metric. That is, if... | 3 | https://mathoverflow.net/users/5340 | 21322 | 14,115 |
https://mathoverflow.net/questions/21257 | 5 | I am interested in examples where the [Shooting Method](http://en.wikipedia.org/wiki/Shooting_method) has been used to find solutions to systems of ordinary differential equations that are either
* reasonably large systems, or
* the search algorithm in the shooting parameters is somewhat prohibitive because of the n... | https://mathoverflow.net/users/3623 | What is state of the art for the Shooting Method? | I don't know about state-of-the-art and I'm not sure if this is the kind of thing you were looking for...however in my two first papers I've used the shooting method in a parameter space that was originally too big (4 and 6 dimensions if I recall correctly) and the problem was that with randomly chosen parameters the n... | 7 | https://mathoverflow.net/users/3578 | 21324 | 14,116 |
https://mathoverflow.net/questions/21298 | 12 | An N-subset $\{x\_1,\dots,x\_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of *Fekete points* (named after [Michael Fekete](http://en.wikipedia.org/wiki/Michael_Fekete)) if it maximizes the product $$\prod\_{1\le k<j\le N}|x\_k-x\_j|\qquad (1)$$ among all such $N$-tuples. When $X\subset \mathbb C$, one c... | https://mathoverflow.net/users/2912 | How to best distribute points on two concentric circles? | (This was too long to fit in a comment.)
For subsets of the complex plane, a slightly different perspective is that the square of the quantity you want to maximize is the absolute value of the discriminant of the monic polynomial whose roots are precisely the points. Up to a factor depending only on the degree, the d... | 2 | https://mathoverflow.net/users/4344 | 21325 | 14,117 |
https://mathoverflow.net/questions/21327 | 4 | It is an easy exercise to show that the Euclidean plane cannot be partitioned into round circles (note however that it is possible to do so for $\mathbb{R}^3$). It seems almost obvious that it is not possible to partition the plane into Jordan curves either.
However, I am not able to design a proof that does not use ... | https://mathoverflow.net/users/4961 | Is it still impossible to partition the plane into Jordan curves without choice? | It is possible to change your argument so that the choice is over countable set; hope this is good enough. Namely, topology on the plane has countable base (say, circles at rational points with rational radii); let's index this base as $U\_1,\dots,U\_n,\dots$; your argument can be used to construct a sequence of Jordan... | 4 | https://mathoverflow.net/users/2653 | 21330 | 14,119 |
https://mathoverflow.net/questions/21328 | 4 | Consider a tree with k nodes and for each node v the vector **l**v = (lv0, lv1, ..., lvk-1) with lvd the number of leaves (!) with distance d to v. I wonder whether two nodes v, w with **l**v = **l**w are [conjugate](https://mathoverflow.net/questions/10807/conjugate-vertices-and-distinguishing-properties) (I guess the... | https://mathoverflow.net/users/2672 | A distinguishing node property in trees? | I have a counterexample. It is not enough just to count leaves, since this doesn't take into account the number of possible ways to arrive at those leaves.
Consider the graph below.
```
A - B - C - D - E - F
| |
G H
|
I
```
I think the vector for C an... | 8 | https://mathoverflow.net/users/1946 | 21335 | 14,123 |
https://mathoverflow.net/questions/21340 | 3 | Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ is free over $A$, and this basis gives a basis of $L/K$.
In particular, if $A=\mathbb{Z}$ or $\mathbb{Z}\_p$ (or even the... | https://mathoverflow.net/users/4399 | Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind. | Watch out: just because $A$ is a PID does not make $B$ a free $A$-module. You need to know that $B$ is finitely generated over $A$ to conclude $B$ has an $A$-basis when $A$ is a PID. If $L/K$ were *separable* then using discriminants you can stuff $B$ inside a finitely generated $A$-module so $B$ is finite free if $A$ ... | 8 | https://mathoverflow.net/users/3272 | 21348 | 14,132 |
https://mathoverflow.net/questions/21287 | 13 | I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
1. There is a proof in Humphreys Linear algebraic groups, he first prove $G/S$ is a complete algebraic variety hence a projective variety, where $S$ is a Borel subgroup of $G$ of larges... | https://mathoverflow.net/users/1851 | How many ways are there to prove flag variety is a projective variety? | I'm not sure this is an answer, but it got too long to be a comment! The projectivity of $G/B$ seems to me to follow from two facts: a) a homogeneous space $G/H$ is always a quasi-projective variety (which is due to Chevalley), and b) the variety $G/B$ must be complete.
The first fact clearly has nothing to do with ... | 7 | https://mathoverflow.net/users/1878 | 21352 | 14,135 |
https://mathoverflow.net/questions/21358 | 1 | Let *C, E* be small categories, let *Ĉ* = Set*C*op, and let *F:Ĉ → Ê* be cocontinuous. I think *F* will always have a right adjoint when *C, E* are small, but not necessarily if they're large. Is that right?
| https://mathoverflow.net/users/756 | Does every cocontinuous functor between categories of presheaves on small categories have a right adjoint? | The part "*F* will always have a right adjoint when *C*, *E* are small" is definitely right. Using some mildly overkill machinery: in this case *Ĉ* and *Ê* are locally presentable categories, and the result is then the **adjoint functor theorem for locally presentable categories**:
Theorem: Let C and D be locally p... | 4 | https://mathoverflow.net/users/126667 | 21363 | 14,143 |
https://mathoverflow.net/questions/21361 | 27 | I was reading recently online Peter May's complaints (I'm a fan, you can tell, I'm sure) about teaching the third quarter of the graduate algebra sequence at the University of Chicago. This course focuses on homological algebra and attempts to be as up-to-date as possible. May's conundrum stems from the fact that homol... | https://mathoverflow.net/users/3546 | The Interrelationship Problem Of Modern Mathematics- How To Deal With it In First Year Graduate Courses? | Of course you should show students, taking into account their backgrounds, that the material they are learning in one course is relevant elsewhere. It makes it clearer to the students that topics they are studying have wide usefulness. At the same time, if you know the students don't have a background to appreciate the... | 15 | https://mathoverflow.net/users/3272 | 21366 | 14,145 |
https://mathoverflow.net/questions/21347 | 2 | Say I have a black box generating data samples, and I want to estimate the parameters of the black box from the samples.
The black box works like this: it has a parameter **m** (a real number), and to generate a value **v**, it first generates v0 according to a normal distribution (with mean *m* and variance 1), and ... | https://mathoverflow.net/users/5363 | Estimating the mean of a truncated gaussian curve | OK, let me fully address the question since there is no easy way out. The
normal approach is to maximize the "likelihood" of the data under the
parameter. The key question here is how to define likelihood for a
mixed distribution. Let's use the standard approach as our guide.
Parameter estimation is usually based on ... | 2 | https://mathoverflow.net/users/4261 | 21371 | 14,148 |
https://mathoverflow.net/questions/21370 | 10 | Basically, I'm aware of "splitting principles" for the following three objects (which are all isomorphic modulo torsion).
**1**. The Chow group a la Fulton.
**2**. The classical Grothendieck group of vector bundles or coherent sheaves.
**3**. The $\gamma$-graded Grothendieck group.
I was just wondering where t... | https://mathoverflow.net/users/4333 | Where does the splitting principle come from and does it generalize | We can think of the splitting principle as a *condition* on a "cohomology theory" (of some sort) $E^\*$, coming about when working with Chern classes for instance, and then ask: When does $E^\*$ satisfy this condition? First, let's make the condition more precise and reformulate it:
**Condition 1:** Given $X$ and a v... | 21 | https://mathoverflow.net/users/1921 | 21381 | 14,154 |
https://mathoverflow.net/questions/21382 | 16 | Let $\mathcal C,\otimes$ be a monoidal category, i.e. $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is a functor, and there's a bit more structure and properties. Suppose that for each $X \in \mathcal C$, the functor $X \otimes - : \mathcal C \to \mathcal C$ has a right adjoint. I will call this adjoint (uniq... | https://mathoverflow.net/users/78 | If "tensor" has an adjoint, is it automatically an "internal Hom"? | It is associative. Consider the evaluation cube drawn [here](http://ejenk.com/misc/internal-hom.pdf). Four of the faces commute by definition of the composition map, and one by functoriality of the tensor product. The commutativity of these five faces implies that any of the maps $W \otimes \operatorname{Hom}(W, X) \ot... | 13 | https://mathoverflow.net/users/396 | 21385 | 14,156 |
https://mathoverflow.net/questions/21367 | 63 | I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those that I read all require the existence of infinitely many primes, which strikes me as extraneous. Is there a known proof that... | https://mathoverflow.net/users/5373 | Proof that pi is transcendental that doesn't use the infinitude of primes | The infinitude of primes (more precisely, the existence of arbitrarily large primes) might actually be necessary to prove the transcendence of $\pi$. As I explained in an [earlier answer](https://mathoverflow.net/questions/19857/has-decidability-got-something-to-do-with-primes/20944#20944), there are structures which s... | 124 | https://mathoverflow.net/users/2000 | 21389 | 14,158 |
https://mathoverflow.net/questions/21373 | 9 | This is a refinement of my (naive, poorly asked) question [here](https://mathoverflow.net/questions/17951/what-tensor-product-of-chain-complexes-satisfies-the-usual-universal-property). The reference for my question is [Baez and Crans, HDA6](http://math.ucr.edu/home/baez/hda6.pdf).
Background: category objects, etc.
... | https://mathoverflow.net/users/78 | What functor is adjoint to the tensor product of 2-vector spaces? | I'll denote your category of 2-vector spaces by 2Vect. By your preliminary remarks, 2Vect is actually the category of Vect-valued presheaves on Δ≤1 where Δ≤1 denotes the full subcategory of Δ on the objects [0] and [1]. Therefore, colimits in 2Vect are computed objectwise under this identification. So the functor – ⊗ X... | 5 | https://mathoverflow.net/users/126667 | 21390 | 14,159 |
https://mathoverflow.net/questions/21386 | 1 | Given $f(0) = 1-c$, what is the non-recursive $f(n)$ that satify the following equation
$f(n)-\frac{1}{2}f(n-1)f(n)+f(n-1) = 1$
for n = 1,2,3,...?
| https://mathoverflow.net/users/5066 | What is the nonrecursive formula for the following implicit function? | $$ f(n) = 2{\frac {1-c+ \left( -1 \right) ^{n}-c \left( -1 \right) ^{n}-\sqrt
{2}c+ \left( -1 \right) ^{n}\sqrt {2}c}{2-\sqrt {2}-\sqrt {2}c+2
\left( -1 \right) ^{n}+ \left( -1 \right) ^{n}\sqrt {2}+ \left( -1
\right) ^{n}\sqrt {2}c}} $$
which is a non-trivial pattern to spot! [I used a CAS] The thing to notice is... | 6 | https://mathoverflow.net/users/3993 | 21391 | 14,160 |
https://mathoverflow.net/questions/21397 | 20 | I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold theory, but may not have had much expose to algebraic geometry. I will loosely follow the book [Introduction to Algebra... | https://mathoverflow.net/users/5337 | What should be taught in a 1st course on Riemann Surfaces? | Good question. I bet you'll get many interesting answers.
About two years ago I taught an "arithmetically inclined" version of the standard course on algebraic curves. I had intended to talk about degenerating families of curves, arithmetic surfaces, semistable reduction and such things, but I ended up spending more ... | 14 | https://mathoverflow.net/users/1149 | 21399 | 14,167 |
https://mathoverflow.net/questions/21383 | 7 | Cauchy's Arm Lemma is used in the proof of Cauchy's Rigidity Theorem for convex polyhedra. The Lemma states that in the plane or on the sphere that if all but one of the side lengths of two convex polygons $P$ and $P'$ are the same, and the angles formed by the remaining sides of P are less than or equal to those of th... | https://mathoverflow.net/users/4490 | Applications of Cauchy's Arm Lemma | The question is a little too general to have a single answer. Cauchy's arm lemma is a basic technical result in rigidity theory. One reason it has a name is because it is intuitively obvious, but the most natural proof Cauchy originally came up with is false.
Now, other than Cauchy theorem, it has few direct conseque... | 7 | https://mathoverflow.net/users/4040 | 21400 | 14,168 |
https://mathoverflow.net/questions/21401 | 23 | I have the opportunity to prepare a research poster for a non-mathematical, yet scientifically savvy audience, and I want to do it well. I have asked a few mathematicians, and I have heard the following sound advice:
* Use interesting graphics.
* Elaborate on possible applications to other scientific fields.
Althou... | https://mathoverflow.net/users/351 | How do you make a good math research poster for a non-mathematical audience? | For technical question 1 I would recommend [beamerposter](http://www-i6.informatik.rwth-aachen.de/~dreuw/latexbeamerposter.php).
| 10 | https://mathoverflow.net/users/290 | 21402 | 14,169 |
https://mathoverflow.net/questions/20593 | 3 | I am trying to understand Horrocks's [construction of vector bundles](http://dl.dropbox.com/u/3849644/Construction%20of%20Bundles.pdf). However I have been stuck on the proof the first theorem in the paper.
In the paper, a trivial bundle is a direct sum of Hopf bundles $\mathcal{O}(p)$.
Theorem: Let $E$ be a vect... | https://mathoverflow.net/users/2348 | Proof of a Theorem in the paper "Construction of bundles on P^n" by Horrocks | That is a pretty terse proof! Let me give an outline of a proof that I know. First, one could deduce the statement from a more general:
**Theorem 1**: Let $R$ be a regular local ring, $E$ be a reflexive $R$-module locally free on $U\_R$, the punctured spectrum such that $E$ has no free direct summand. Then one can fi... | 4 | https://mathoverflow.net/users/2083 | 21420 | 14,181 |
https://mathoverflow.net/questions/21415 | 44 | Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped with a faithful exact functor "forget" to the category of finite-dimensional vector spaces over $\mathbb C$. Moreover, $\m... | https://mathoverflow.net/users/78 | What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra? | Rather than an answer, this is more of an anti-answer: I'll try to persuade you that you probably don't want to know the answer to your question.
Instead of some exotic nonalgebraic Lie algebra, let's start with the one-dimensional Lie algebra $\mathfrak{g}$ over a field $k$. A representation of $\mathfrak{g}$ is jus... | 31 | https://mathoverflow.net/users/930 | 21429 | 14,188 |
https://mathoverflow.net/questions/21245 | 24 | The Erdős-Ko-Rado theorem talks about how large an intersecting set system (a set of pairwise intersecting sets) can be if the size of the base set is fixed. I'm interested about intersecting set systems where the base set is not fixed, but the size of the sets is bounded. I can prove the following lemma (see proof bel... | https://mathoverflow.net/users/5340 | Pairwise intersecting sets of fixed size | This isn't an answer but simply a way of thinking about the question that I quite like. (Later: see below for an attempted proof.)
Let's regard each set in our collection as a vertex of a graph, and let's join two vertices if and only if the corresponding sets intersect. That doesn't sound very interesting, since we ... | 10 | https://mathoverflow.net/users/1459 | 21435 | 14,194 |
https://mathoverflow.net/questions/20890 | 3 | Recall that a morphism of rings $R\to S$ is called (essentially) *smooth* if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is *essentially finitely presented* provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$,... | https://mathoverflow.net/users/1353 | Lifting results from smooth maps to essentially smooth maps. | It seems that what you are looking for is theorem 5.11 [here](http://arxiv.org/abs/0809.1201). See also example (e) on section 5.12. Also if you don't feel like reviewing from EGA you can look at section 1.5 of "Introduction to algebraic stacks" by A. Canonaco which I think covers the relevant facts (including 17.5.1)
... | 4 | https://mathoverflow.net/users/2384 | 21436 | 14,195 |
https://mathoverflow.net/questions/20915 | 1 | Let $X$ be a scheme. Consider the preorder of locally closed immersions into $X$ (considered, say, as a full subcategory of $Sch/X$). Is it complete or cocomplete? That is, are there infima or suprema?
Ok it's easy to see that every nontrivial locally closed subscheme of $\mathbb{A}^1\_\mathbb{C}$ contains only fini... | https://mathoverflow.net/users/2841 | is the preorder of locally closed immersions complete? | The maximum may not exist in general. Take X=Spec k[T,U] the affine plane, A the complement of the vertical line L passing through the origin (A=Spec k[T,U,1/T]) and B the origin (Spec k[T,U]/(T,U)). Then, the maximum C of A and B in the ordered set of subschemes of X does not exist. If it was the case, then C should b... | 3 | https://mathoverflow.net/users/5383 | 21450 | 14,206 |
https://mathoverflow.net/questions/21424 | 40 | I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers.
Can someone please provide some hints on what can be used, preferably with some example codes? I do not need complicated di... | https://mathoverflow.net/users/2083 | How to draw knots with Latex? | Knotinfo has .png files of all knots of 12 crossings or less.
<http://www.indiana.edu/~knotinfo>
| 46 | https://mathoverflow.net/users/3874 | 21456 | 14,209 |
https://mathoverflow.net/questions/21458 | 6 | Prompted by this [question](https://mathoverflow.net/questions/21424/how-to-draw-knots-with-latex) I would like to ask the community how they convert their mathematics into pdf files. In any given procedure for converting mathematics into pdf I am interested in two issues: first typographical quality of text and of mat... | https://mathoverflow.net/users/1650 | Typesetting mathematics: how do {\em you} convert text into pdf? | You can do margin kerning (aka “protrusion”) and font expansion in pdfLaTeX simply by loading the package `microtype` (i.e., by adding
>
> `\usepackage{microtype}`
>
>
>
to the preamble). I also suggest using the `tracking` option for small-caps, which increases the space between letters (which is typographica... | 8 | https://mathoverflow.net/users/5304 | 21466 | 14,214 |
https://mathoverflow.net/questions/21444 | 5 | Let $G$ be the reductive group $\operatorname{GSp}\_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}\_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for caution, that $\pi$ satisfies a multiplicity one hypothesis.
Fix $p$ an odd prime. To $\pi$ is attached a $p$-adic repr... | https://mathoverflow.net/users/2284 | What is the image of complex conjugation under Siegel Galois representations? | The preprint "Conjecture de type de Serre et formes compagnons pour $GSp\_4$" by Florian Herzig and Tilouine (available [here](http://math.northwestern.edu/~herzig/ht08-7c.pdf)) indicates that there should be two 1's and two -1's.
EDIT: Having come in to work today, I could look up the more precise statement given in... | 5 | https://mathoverflow.net/users/1021 | 21469 | 14,217 |
https://mathoverflow.net/questions/21471 | 7 | When I was at school I wondered if a surface could locally appear to be a unit sphere, yet `carry on forever'. More formally, my question is:
Can you place a metric of constant curvature +1 on ${\mathbb R}^2$, such that the identity map to ${\mathbb R}^2$ (with standard Euclidean metric) is uniformly continuous?
It... | https://mathoverflow.net/users/5394 | Can you have a spherical plane? | By Myers's theorem (see <http://en.wikipedia.org/wiki/Myers%27s_theorem>) you have that if the Ricci curvature of a complete $n$-manifold $M$ is bounded below by $(n − 1)k > 0$, then its diameter is bounded by some constant depending on $k$. In particular, it is compact. Therefore if there is such a metric, it cannot b... | 10 | https://mathoverflow.net/users/3995 | 21474 | 14,219 |
https://mathoverflow.net/questions/21470 | 12 | In a comment to [this question](https://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice), Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to be shown that choice is... | https://mathoverflow.net/users/2233 | Subset of the plane that intersects every line exactly twice | By AC, choose a *cardinal* well-ordering of the lines in in the plane and any well-ordering of all the points.
We proceed by transfinite induction.
Suppose $A\_l$ is a set of points, no three colinear, and let $B\_l$ be the set of lines spanned by points of $A\_l$, and let $C\_l=\cup B\_l$. Suppose further that $l'... | 8 | https://mathoverflow.net/users/1631 | 21476 | 14,221 |
https://mathoverflow.net/questions/21473 | 6 | If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ has $k$ distinct elements in it? In other words:
>
> Must every matrix of finite multiplicative order have a regular... | https://mathoverflow.net/users/3710 | Linear algebra and regular orbits | For your first question, I presume you also wish to insist that $k$
be the *least* integer such that $A^k=I$. The matrix $A$ is then similar
over your field to a direct sum $B\_1,\ldots,B\_m$ of companion matrices
of (over your field $F$) factors of $X^k-1$, say $f\_1,\ldots,f\_m$.
Then $F^n$ decomposes as a direct sum... | 4 | https://mathoverflow.net/users/4213 | 21477 | 14,222 |
https://mathoverflow.net/questions/21484 | 14 | The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.
Let $A$ be a local Cohen-Macaulay ring of dimension $d$.
1. Let $I$ be an ideal generated by $r$ elements. Is is true that the depth of $A/I$ is at least $d-r$?
2. Let $Q$ be a minimal prime of... | https://mathoverflow.net/users/1594 | Two questions about Cohen-Macaulay rings | Nice questions! The answers are no in both cases, although the examples are more interesting than one would expect.
1) Even when $A$ is regular, one can always find an ideal $I$ with $3$ generators such that $A/I$ has depth $0$. This is due to a very nice [result](http://www.ams.org/mathscinet/search/publdoc.html?pg1... | 18 | https://mathoverflow.net/users/2083 | 21488 | 14,228 |
https://mathoverflow.net/questions/21492 | 11 | I have two naive questions about stacks.
>
> 1) Is it possible to define stacks in the Zariski topology?
>
>
>
Presuming you can:
>
> 2) If a stack has a coarse moduli, and the coarse moduli space is a scheme, then does that mean that your stack is a stack in the Zariski topology?
>
>
>
In general, I... | https://mathoverflow.net/users/1231 | Stacks in the Zariski topology? | 1.) It's possible to define stacks on ANY category equipped with a Grothendieck topology (such a category with a topology is called a site). In particular, this holds true for the Zariski site. Moreover, there is always a way to define an "Artin stack"- these are those stacks which arise as torsors for a groupoid objec... | 11 | https://mathoverflow.net/users/4528 | 21500 | 14,234 |
https://mathoverflow.net/questions/21427 | 5 | I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
***Preamble***
We consider polynomials f ∈ ℤ[x] with roots in ℝ, and for each polynomial f, the **principal root** is the real root with the largest magnitude... | https://mathoverflow.net/users/3723 | Transformations of integer polynomials under combinations of their roots | If $u$ is a root of $f$ and $v$ is a root of $g$ then $u+v$ is a root of the resultant of $f(x-y)$ and $g(y)$ (for the purpose of calculating the resultant, we take these as polynomials in $y$). The resultant is just a determinant of a matrix whose entries are all coefficients of the polynomials, so it would seem to sa... | 3 | https://mathoverflow.net/users/3684 | 21519 | 14,245 |
https://mathoverflow.net/questions/21513 | 19 | Let $K$ be a number field and let $\mathcal O\_K$ be the ring of integers. Following [this paper](https://www.emis.de/journals/JTNB/2005-3/article02.pdf "Division-ample sets and the Diophantine problem for rings of integers") of Cornelissen, Pheidas, and Zahidi, a key ingredient needed to show that Hilbert's tenth prob... | https://mathoverflow.net/users/4872 | Can you show rank E(Q) = 1 exactly for infinitely many elliptic curves E over Q without using BSD? | I don't think that this should be too hard: take a simple family of curves, such as
$y^2 = x^3 + px$ or something similar, and choose $p$ from a certain set of residue classes to guarantee that the 2-Selmer group has rank 1. You can complete the proof either by invoking rather deep constructions using Heegner points, ... | 16 | https://mathoverflow.net/users/3503 | 21534 | 14,254 |
https://mathoverflow.net/questions/21483 | 5 | I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The associated spectral sequence is highly degenerate. Of course, in degenerate cases, one hopes that there are techniques for get... | https://mathoverflow.net/users/5395 | Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves | Ok, so what will the spectral sequence give you? This is a very easy exercise, but since Altgr is not experienced, here is the solution. The term $E\_2^{p,q}$ is the $p^{\rm th}$ cohomology group of the complex $\mathrm{H}^q K^{\bullet}$ (the complex of groups obtained by applying the $q^{\rm th}$ cohomology functor to... | 7 | https://mathoverflow.net/users/4790 | 21536 | 14,256 |
https://mathoverflow.net/questions/21512 | 17 | A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for complete boundaryless Alexandrov spaces (of curvature bounded below)?
Alexandrov spaces are manifolds almost everywhe... | https://mathoverflow.net/users/4354 | Is there Domain Invariance for Alexandrov spaces? | The following lemma from *[Grove--Petersen, A radius sphere theorem](http://www.math.psu.edu/petrunin/papers/alexandrov/radius_sphere_theorem+.pdf)* does the trick.
>
> **Lemma 1.** Let $X$ be a compact Alexandrov space without boundary. Then $X$ has a fundamental class in Alexander-Spanier cohomology with $\mathbb... | 7 | https://mathoverflow.net/users/1441 | 21537 | 14,257 |
https://mathoverflow.net/questions/6764 | 10 | Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless I've made a mistake, this is always be an fpqc sheaf. When $M$ is a finitely generated free A-module, then $\mathrm{Spe... | https://mathoverflow.net/users/788 | When is tensoring with a module representable by a scheme? | When $A$ is noetherian and $M$ is finitely generated, Nitin Nitsure showed that the functor is representable if and only if $M$ is projective (see <http://arxiv.org/abs/math/0308036>).
| 4 | https://mathoverflow.net/users/4790 | 21539 | 14,258 |
https://mathoverflow.net/questions/21521 | 8 | The Haag-Kastler approach to quantum field theory (QFT) is one of the oldest approaches to rigorously define what a QFT is, it deals with nets of operator algebras: You start with a spacetime and assign von Neumann algebras (or $C^\*$-algebras, but my question is about the von Neumann algebra situation only) to certain... | https://mathoverflow.net/users/1478 | Murray-von Neumann classification of local algebras in Haag-Kastler QFT | There is a nice overview about algebraic quantum field theory by Halverson and Müger, which covers some of the stuff I mention below and can be found at
<http://arxiv.org/abs/math-ph/0602036>
Concerning your question(s): Having a factor of type III means that every (non-zero) projection in it is Murray-von Neumann ... | 6 | https://mathoverflow.net/users/3995 | 21551 | 14,264 |
https://mathoverflow.net/questions/21555 | 16 | Modular forms could be defined for arbitrary subgroups of the modular group, and I have read that this is done in some papers, but every definition of a modular form I have seen has been with respect to congruence subgroups.
| https://mathoverflow.net/users/4692 | Why are modular forms (usually) defined only for congruence subgroups? | At a certain level, it's mostly a matter of (i) terminology and (ii) reading the right books. Technically the word "modular" in modular forms refers to the "modular group $SL\_2(\mathbb{Z})$".
In Miyake's book *Modular Forms*, he defines an **automorphic form** with respect to an arbitrary Fuchsian group $\Gamma$ (i.... | 22 | https://mathoverflow.net/users/1149 | 21556 | 14,267 |
https://mathoverflow.net/questions/21565 | 2 | In "Infinite dimensional analysis, A hitchhikers guide" by Aliprantis and Border, they write that these 2 classes of topologies "by and large include everything of interest".
@Pete Clarke: I was asking if it holds for mathematics in general, and not just for functional analysis. From the answers I get the impression ... | https://mathoverflow.net/users/4692 | Is it true that the only interesting topologies are metric topologies and weak topologies? | Picking up on Gerald's interpretation of the question (namely, that it really focusses on infinite dimensional vector spaces) then I say: absolutely not!
For example, piecewise-smooth paths in some Euclidean space has a topology that is neither of these (it's an uncountable inductive limit of Frechet spaces). (Not th... | 13 | https://mathoverflow.net/users/45 | 21569 | 14,277 |
https://mathoverflow.net/questions/21527 | 7 | **Background:** It is possible (see e.g., [this](http://www.cs.cmu.edu/%7Elblum/PAPERS/TuringMeetsNewton.pdf)) to define a Turing machine over an arbitrary ring. It reduces to the classical notion when the ring is $\mathbb{Z}\_2$; the key difference is that elementary algebraic computations are allowed to be performed ... | https://mathoverflow.net/users/344 | Does IP = PSPACE work over other rings? | I think PSPACE does not make much sense over arbitrary rings (it is better to consider classes such as PAR).
Consider $R=\mathbb Z$ or $R=\mathbb Q$ with order. You can encode any finite collection of numbers in a single number and extract elements back in constant space. So you can emulate a potentially infinite mem... | 5 | https://mathoverflow.net/users/4354 | 21573 | 14,281 |
https://mathoverflow.net/questions/21313 | 8 | [[UPDATE: This work has now been published at SIAM J Discrete Math.: [Formulae for the Alon–Tarsi Conjecture](http://epubs.siam.org/sidma/resource/1/sjdmec/v26/i1/p65_s1).]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one unpublished formula) for the number of even Latin square... | https://mathoverflow.net/users/2264 | (0,1)-matrix congruence: is it known? | Darij's proof was wrong but both his claims (that it is next to trivial and that it requires some long formulae) were actually correct. Here is the demonstration.
To start with, we'll count the sum over degenerate (in $\mathbb Z\_p$) matrices instead because the sum over all matrices is clearly $0$. Also, I'll prefer... | 4 | https://mathoverflow.net/users/1131 | 21574 | 14,282 |
https://mathoverflow.net/questions/21552 | 48 | I have read Hartshorne's *Algebraic Geometry* from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry.
* I want to know how to use scheme theory and its cohomology to solve arithmetic problems.
* I also want to learn something about moduli theory.
Would yo... | https://mathoverflow.net/users/5274 | Roadmap for studying arithmetic geometry | My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.
One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tat... | 29 | https://mathoverflow.net/users/2874 | 21583 | 14,289 |
https://mathoverflow.net/questions/21578 | 36 | I proposed to a master's student to work, from the exercise in Ghys-de la Harpe's book, on the proof that a finitely generated group $G$ that is quasi-isometric to $\mathbb{Z}$ is virtually $\mathbb{Z}$. However I initially had in mind the result that gives the same conclusion from the hypothesis that $G$ has linear gr... | https://mathoverflow.net/users/4961 | Is there a simple proof that a group of linear growth is quasi-isometric to Z? | Here are two papers dealing with this question:
Wilkie, A. J.(4-MANC); van den Dries, L.(1-IA-S)
An effective bound for groups of linear growth.
Arch. Math. (Basel) 42 (1984), no. 5, 391--396.
20F05
and
Imrich, Wilfried(A-MNT); Seifter, Norbert(A-MNT)
A bound for groups of linear growth.
Arch. Math. (Basel) 48 (19... | 10 | https://mathoverflow.net/users/3380 | 21586 | 14,291 |
https://mathoverflow.net/questions/20730 | 5 | Preliminars and notation:
Let $M$ be an $n$-dimensional compact manifold, $T\colon M\rightarrow M$ a diffeomorphism and $( x\_n)\_{n\in\mathbb{Z}}$ a dense orbit under $T$, ($x\_n = T^n(x\_0)$). Let $p\in M$ be another point and define, for $\delta>0$, $B\_n(\delta) = B(p, e^{-n\delta})$.
Question: Is it true that... | https://mathoverflow.net/users/5231 | About dense orbits on dynamical systems | This is called the *shrinking target problem*, and there is a reasonably large literature on it. For hyperbolic dynamical systems we can usually find quite a few pairs $x$, $p$ such that $A$ is infinite for all $\delta$. Indeed, I believe that there are results showing that in certain cases, for any point $z$ and posit... | 6 | https://mathoverflow.net/users/1840 | 21588 | 14,292 |
https://mathoverflow.net/questions/21232 | 22 | Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}\_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-singular subvariety is well-known to be non-singular, so the real question is ``what happens when $Z$ is singular?" The b... | https://mathoverflow.net/users/5337 | When is a blow-up non-singular? | Craig Huneke told me about this [paper](https://www.tandfonline.com/doi/abs/10.1080/00927879708825958): "On the smoothness of blow-ups" ([MR1446135](https://mathscinet.ams.org/mathscinet-getitem?mr=1446135), [Zbl 0878.13004](https://zbmath.org/?q=an%3A0878.13004), by O'Carroll and Valla). The title alone seems to sugge... | 12 | https://mathoverflow.net/users/2083 | 21592 | 14,294 |
https://mathoverflow.net/questions/21591 | 5 | Here is the essence of a problem I have run in to: I have a finite poset D with a terminal object. If I formally invert all of the morphisms, and add these into my diagram, does the new diagram D' still commute?
I think that the resulting diagram will still commute basically because I have done a lot of examples. Wor... | https://mathoverflow.net/users/1106 | When does adding inverses of morphisms preserve commutativity of a diagram? | There is an easy conceptual proof using the fact that the category obtained by formally inverting all the arrows in a category C is equivalent to the fundamental groupoid of the nerve NC of C, and that the nerve of a category with a final object is contractible. Without the assumption of a final object your assertion i... | 5 | https://mathoverflow.net/users/126667 | 21596 | 14,296 |
https://mathoverflow.net/questions/21599 | 35 | Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this compactification (or equivalently, $x$ and $y$ are two non-principal ultrafilters). I am interested in ways to "model" the ultrafi... | https://mathoverflow.net/users/766 | "Transitivity" of the Stone-Cech compactification | The answer to Q1 is no. This has been well studied in set theory; you're basically asking whether any two non-principal ultrafilters on $\mathbb{N}$ are comparable under the Rudin-Keisler ordering. Variations on your question have led to many, many interesting developments in set theory, but your question Q1 is easy to... | 43 | https://mathoverflow.net/users/2000 | 21605 | 14,299 |
https://mathoverflow.net/questions/21604 | 11 | I am writing something for the journal of the university on the Lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \cdot v=-1$ is a metric:
*The hyperbolic distance between $u$ and $v$ is the only positive number $\eta (u,v)$ such that*... | https://mathoverflow.net/users/1619 | Nice proof of the triangle inequality for the metric of the hyperbolic plane | We need to check $\eta(u,v)+\eta(v,w)\ge\eta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$.
First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an isometry, we may now assume that $v=(0,0,1)$. This is the main idea. For added convenience, you may also rotate the $x... | 12 | https://mathoverflow.net/users/4354 | 21608 | 14,300 |
https://mathoverflow.net/questions/21601 | 3 | This is version 2 of a question about the ultimate limits of Tennenbaum's Theorem. The attempt to find these limits by moving up the induction heirarchy, as in Wilmer's Theorem, seems somehow indecisive. I suggested that maybe there is a Theory $T$ extending open induction such that
1) $T$ has a recursively presentab... | https://mathoverflow.net/users/5229 | Ultimate limits of Tennenbaum's Theorem | This is equivalent to saying that $T$ is complete. Let $M$ be a recursively presentable model of $T$. If $T$ is not complete, then we can find $\phi$ which is not provable from $T$ but is true in $M$. Of course, $M$ is then a recursively presentable model of $T+\phi$. If $T$ is complete then $T+\phi$ has no model at al... | 4 | https://mathoverflow.net/users/2000 | 21609 | 14,301 |
https://mathoverflow.net/questions/21614 | 3 | My question is coming from the method Reid and Chris suggested in solving the problem [here](https://mathoverflow.net/questions/19116/colimits-in-the-category-of-smooth-manifolds). Help on any point is greatly appreciated!
Question 1. For a real manifold $M$, consider $C^{\infty}(M,\mathbb{R})$. For a point $p\in M$... | https://mathoverflow.net/users/4517 | Ideals in the ring of smooth endomorphisms of the real line | This is all fairly well-known stuff. In Q1, the ideal $I\_p^n$ is the
set of functions for which the derivatives up to order $n-1$
at $p$ vanish in a local coordinate system centred at $p$.
I'll just give a rough outline of how to reduce to the case where $M=\mathbb{R}^k$.
Take a coordinate neighbourhood $U$ of $p$ w... | 4 | https://mathoverflow.net/users/4213 | 21615 | 14,305 |
https://mathoverflow.net/questions/21616 | 2 | Let A be a pair (i, x)
$H(A)$ <=> program i halts on input x
$P(A)$ <=> (there exists a proof for $H(A)$) $\vee$ (there exists a proof for $\neg H(A)$)
Assume $\forall A: P(A)$ then we can solve the halting problem (write an algorithm that enumerates all proofs and checks whether they proof $H(A)$ or $\neg H(A)$)... | https://mathoverflow.net/users/5415 | Provability of termination. Whats wrong with my reasoning? | I believe that the error sneaks in with the phrase "for such an A." It is indeed true that for any $A$ such that $\neg P(A)$ we have that $\neg H(A)$; it follows that to be able to prove $\neg P(A)$ actually requires us to be able to prove $\neg H(A)$. You've substituted the *a priori* knowledge "this is such an $A$" f... | 5 | https://mathoverflow.net/users/4658 | 21618 | 14,307 |
https://mathoverflow.net/questions/21562 | 29 | Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work?
The only example I know is Cantor's theory of sets. Nothing like his concrete manipulations of actual infinite objects had been done before.
... | https://mathoverflow.net/users/4692 | What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work? | Shannon's work on Information theory. Maybe the math wasn't new but the ideas (such as positing a qualitative metric of information and identifying its relevance to design of communication systems) definitely were.
| 36 | https://mathoverflow.net/users/2878 | 21620 | 14,308 |
https://mathoverflow.net/questions/21543 | 8 | Hi,
I have what I hope is a very simple question related to unfamiliar notation.
I am looking through a maths paper on a topic related to set theory which contains a symbol,
$\uplus$,
and I would like to know how, if at all, it differs from the typical
$\cup$
symbol in terms of its meaning.
The contex... | https://mathoverflow.net/users/179 | What is the definition of the $\uplus$ symbol? | For the sake of giving a response to tag as an answer:
Some authors use this to denote a disjoint union of sets, i.e., "$A \cup B$ for sets $A, B$ where $A \cap B = \emptyset$."
| 11 | https://mathoverflow.net/users/572 | 21644 | 14,319 |
https://mathoverflow.net/questions/21641 | 11 | What is the maximal number of perfect matchings a graph $G(V,E)$ can have if $|V|$ and $|E|$ are fixed? I am particularly interested in a case when $|E| = c|V|^2$.
| https://mathoverflow.net/users/2641 | Maximum number of perfect matchings in a graph | I think this is exactly the main result of [this recent paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-4X9D5FW-2&_user=4423&_coverDate=02%2F28%2F2010&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1298182872&_rerunOrigin=scholar.google&_acct=C000059605&_version=1&_urlVersi... | 13 | https://mathoverflow.net/users/4040 | 21648 | 14,323 |
https://mathoverflow.net/questions/21651 | 33 | For $K$ a compact Lie-group with maximal torus $T$, I'd like to know the cohomology $\text{H}^{\ast}(K/T)$ of the flag variety $K/T$.
If I'm not mistaken, this should be isomorphic to the algebra of coinvariants of the associated root system, according to the "classical Borel picture", as it is often called in the li... | https://mathoverflow.net/users/3108 | Cohomology of Flag Varieties | Borel's lengthy 1953 Annals paper is essentially his 1952 Paris thesis. It was
followed by work of Bott, Samelson, Kostant, and others, which eventually answers your
side question affirmatively. For a readable modern account in the setting
of complex algebraic groups rather than compact groups, try to locate a copy of ... | 21 | https://mathoverflow.net/users/4231 | 21659 | 14,328 |
https://mathoverflow.net/questions/21656 | 2 | I know this might be a very elementary question. But I could not find the original definition of Extreme(or Extremal)vectors of some weights $\lambda$(fixed the $w\in W$,where $W$ is Weyl group)
I am looking for definition for Extreme vector for finite dimensional Lie algebra and Affine Lie algebra. I found a paper s... | https://mathoverflow.net/users/1851 | What is Extreme/Extremal vector according to some weights | Usually "extremal weight" means a weight in the Weyl group orbit of the highest, and I would interpret "extremal vector" as an element of said weight space.
| 7 | https://mathoverflow.net/users/66 | 21662 | 14,331 |
https://mathoverflow.net/questions/21667 | 41 | This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: **"Is the homotopy category of topological spaces a concrete category (in the sense, say, of Kurosh and Freyd)?"**. You may s... | https://mathoverflow.net/users/4721 | Are there any "homotopical spaces"? | [No](http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html).
| 59 | https://mathoverflow.net/users/290 | 21668 | 14,332 |
https://mathoverflow.net/questions/21639 | 3 | Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least available for people without PhD in AG) the better.
| https://mathoverflow.net/users/5419 | Amazing examples in complex Algebraic Geometry | I will give an example which I was looking for a long time by myself. The question concerns CONTRACTIBILITY OF CURVES ON SMOOTH SURFACES:
Having a smooth (hence projective) surface $X$ and a divisor $E$ when can $E$ be contracted into an algebraic singularity (i.e. such that the quotient $X/E$ is an algebraic surface)?... | 4 | https://mathoverflow.net/users/5419 | 21678 | 14,337 |
https://mathoverflow.net/questions/21627 | 3 | This question arose from the responses to [this question.](https://mathoverflow.net/questions/21232/when-is-a-blow-up-non-singular) The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the closed subscheme $Z$ defined by $(x,y)^2$ is non-... | https://mathoverflow.net/users/5337 | What are non-trivial examples of non-singular blow-ups of a non-singular variety? | Suppose that $X = \mathbb{A}^2$. Let $Y$ be the blow up of $X$ at the maximal ideal $(x,y)$ and let $W$ be the blow up of $Y$ at a point on the exceptional divisor of $Y$ over $X$. Of course, the composition $f: W \rightarrow X$ is birational and an isomorphism away from the origin. The fiber of $f$ over the origin is ... | 5 | https://mathoverflow.net/users/397 | 21679 | 14,338 |
https://mathoverflow.net/questions/21682 | 14 | I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.
Is it true? Where can I find a proof?
A counterexample for open subsets of $\mathbb{R}^n$ is furnished by the halfplane with the $ax+b$ law.
| https://mathoverflow.net/users/1049 | Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent? | This is true and is in "Michel Lazard: Sur la nilpotence de certains groupes algébriques, Comptes Rendus, vol 241, 1955, 1687--1689"
| 15 | https://mathoverflow.net/users/4008 | 21689 | 14,342 |
https://mathoverflow.net/questions/21683 | 1 | This question follows the article discussed [here](https://mathoverflow.net/questions/16764/equality-of-the-sum-of-powers)
---
Problem
-------
Suppose we're trying to bound the number of integral solutions to a system of multi-variable polynomials,
say
$$ \sum\_{i=1}^n x\_i^t = \sum\_{i=1}^n y\_i^t, $$
where ... | https://mathoverflow.net/users/4248 | Number of integral solutions to multi-variable polynomials | People studying Waring's problem via the Hardy-Littlewood method often consider this kind of problem. You could start by looking at Vaughan's book, "The Hardy-Littlewood method".
| 3 | https://mathoverflow.net/users/2290 | 21694 | 14,346 |
https://mathoverflow.net/questions/20418 | 8 | In developing a theory of index for inclusions of finite von Neumann algebras, several authors ([Kosaki, 1986], [Fidaleo & Isola,1996], etc.) define the index of a conditional expectation of a von Neumann algebra M onto a vN-subalgebra N (here, a conditional expectation is a normal, faithful N-N bimodule map fixing the... | https://mathoverflow.net/users/2085 | When does a conditional expectation preserve some trace? | I would expect a general answer to be difficult, because the set of traces on your von Neumann algebra will depend a lot on the centre of the algebra.
In the case of a factor, the question becomes how to tell if a given expectation is the one that commutes with the trace. Not checking all my facts very carefully, I ... | 3 | https://mathoverflow.net/users/3698 | 21712 | 14,358 |
https://mathoverflow.net/questions/21707 | 4 | Consider an Ab-category $\mathcal{D}$ and its pseudo-abelian envelope $\mathcal{C}$. If necessary, one can assume $\mathcal{D}$ to be $k$-linear over a field $k$ and have finite-dimensional hom-spaces. I am interested in conditions on $\mathcal{D}$ that will ensure that $\mathcal{C}$ is abelian. Are there any such nont... | https://mathoverflow.net/users/344 | When is the pseudo-abelian envelope abelian? | A sufficient (but not necessary) condition is that the endomorphism ring of every object be semi-simple. See [Jannsen](http://www.ams.org/mathscinet-getitem?mr=1150598).
| 6 | https://mathoverflow.net/users/5434 | 21714 | 14,360 |
https://mathoverflow.net/questions/21709 | 21 | For example, I find the first group isomorphism theorem to be vastly more opaque when presented in terms of commutative diagrams and I've had similar experiences with other elementary results being expressed in terms of exact sequences. What are the benefits that I am not seeing?
| https://mathoverflow.net/users/4692 | What are the advantages of phrasing results in terms of exact sequences and commutative diagrams? | Holy cow, go *beyond* the first homomorphism theorem! For example, if you have a long exact sequence of vector spaces and linear maps
$$
0 \rightarrow V\_1 \rightarrow V\_2 \rightarrow \cdots \rightarrow V\_n \rightarrow 0
$$
then exactness implies that the alternating sum of the dimensions is 0.
This generalizes the ... | 62 | https://mathoverflow.net/users/3272 | 21715 | 14,361 |
https://mathoverflow.net/questions/21664 | 5 | I am trying to find out what results are already out there in this direction.
For example, from the Ito-Michler theorem, if $\rho(G)$ is the set of prime divisors of irreducible characters of $G$ and either $\rho(G)\subseteq \pi$ or $|\rho(G)\cap\pi|\leq 1$ then we are guaranteed the existence of a $\pi$-Hall subgroup.... | https://mathoverflow.net/users/4614 | Existence of certain Hall-subgroups based on knowledge of the degrees of irreducible characters | Background:
>
> (Ito–Michler) p does not divide the order of the degree of any (absolutely) irreducible (ordinary) character of G iff G has a normal, abelian, Sylow p-subgroup.
>
>
>
For any subset π of ρ(G)′, it follows that G has a normal, abelian, Hall π-subgroup, and hence also a Hall π′-subgroup. Flipping... | 7 | https://mathoverflow.net/users/3710 | 21716 | 14,362 |
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