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https://mathoverflow.net/questions/85492 | 14 | The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.
>
> **Question.** Let $X$ be a Banach space, and let
> $V$ and $W$ be complemented subspaces
> of $X$. Is it true that $V \cap W$
> is a complement... | https://mathoverflow.net/users/6101 | Intersection of complemented subspaces of a Banach space | The answer to the first question is "no". You can see this with specific examples, but here is a more conceptual approach: Take $Y$ an uncomplemented subspace of $X$ and in $Z:= X\oplus\_1 X$ identify $Y\oplus 0$ with $0 \oplus Y$ in the obvious way; that is, mod out from $Z$ the subspace $\{(y,-y) | y \in Y\}$. $X\opl... | 16 | https://mathoverflow.net/users/2554 | 85557 | 50,931 |
https://mathoverflow.net/questions/85561 | 2 | Let $X$ be an algebraic variety. Let $\mathcal I\_{\Delta}\subset\mathcal O\_{X\times\_kX}$ be the ideal sheaf defining the diagonal $\Delta\subset X\times\_kX$. Regard $\mathcal O\_{X\times\_kX}/\mathcal I^{n+1}$ and $\mathcal I/\mathcal I^{n+1}$ as $\mathcal O\_X$-modules through the first projection.
The question... | https://mathoverflow.net/users/20544 | Are sheaves of principal parts locally free over some dense open set? | They are locally free of constant rank over the smooth locus $U$ of $X$, which is dense and open if $X$ is reduced and irreducible. (I have taken the phrase "variety over a field $k$" to mean: of finite type over $k$, reduced and absolutely irreducible.) The reason, starting from $n=1$, is that $\mathcal O\_{X\times X}... | 8 | https://mathoverflow.net/users/8726 | 85566 | 50,936 |
https://mathoverflow.net/questions/85569 | 5 | Let $\Gamma$ be a cofinite lattice in $PSL(2,\mathbb{R})$ with torsion subgroup $H$.
Is the a uniform bound on the cardinality of $H$?
| https://mathoverflow.net/users/10400 | Is the cardinality of occuring torsion subgroups in cofinite lattices in SL(2,R) bounded? | There are triangle groups $(2, 3, n)$ for any $n>6,$ so I would say the answer is **NO**
| 6 | https://mathoverflow.net/users/11142 | 85571 | 50,938 |
https://mathoverflow.net/questions/85579 | 2 | I will introduce my question with the following example:
For the partition of 26 = 1+1+1+2+3+3+5+5+5 let us calculate its "fancy number" as follows:
From the group of the ones, we get: 1\*(1+1)*(1+1+1)=6
From the group of the twos (actually only one), we get: 2
From the group of the thress, we get: 3*(3+3)=18
From ... | https://mathoverflow.net/users/20506 | a type of numbers resulting from the partitions of an integer | First of all, let's formalize the definition: Let $\lambda$ be a partition. For every integer $i \geq 1$, let $m\_i\left(\lambda\right)$ be the number of appearances of $i$ in $\lambda$. Then your fancy number is $\prod\limits\_{i\geq 1} \left(m\_i\left(\lambda\right)! i^{m\_i\left(\lambda\right)}\right)$. This number ... | 12 | https://mathoverflow.net/users/2530 | 85581 | 50,942 |
https://mathoverflow.net/questions/84636 | 2 | It is known that the *codomain fibration* is given by a functor in the form $\mathcal{C}^{\rightarrow}\longrightarrow\mathcal{C}$ where $\mathcal{C}$ is a category having pullbacks and $\mathcal{C}^{\rightarrow}$ is the *arrow category*. It is also known (see B. Jacobs, *Categorical Logic and Type Theory*, Studies in L... | https://mathoverflow.net/users/3338 | Codomain fibration. | I assume, like Finn, that by "cartesian" you mean "having finite products". It's not really clear to me what you're looking for, since you said yourself that a necessary and sufficient condition for the codomain functor to be a fibration is that $\mathbf{C}$ have pullbacks.
There is however a vaguely "codomain-like" ... | 1 | https://mathoverflow.net/users/49 | 85597 | 50,947 |
https://mathoverflow.net/questions/39874 | 4 | First of all, congratulations to [Dömötör](http://arxiv.org/abs/1009.4641)! This question is related to an interesting [problem](https://mathoverflow.net/questions/26358/can-we-color-z-with-n-colors-such-that-a-2a-na-all-have-different-colors) he asked a while ago. (And it is an attempt to have people take a new look a... | https://mathoverflow.net/users/6085 | Non-multiplicative 6-colorings of Z^+ | First of all, thank you! (Sorry I am reading this a little late, but I have not noticed this question before...) Let me also congratulate your student! (To whom small note: on his slide 34 (from <http://caicedoteaching.files.wordpress.com/2012/01/presentation-7.pdf>) the modulos seem to be mistyped.)
Regarding your q... | 1 | https://mathoverflow.net/users/955 | 85602 | 50,949 |
https://mathoverflow.net/questions/85562 | 14 | I have spent a lot of time trying to track down the following without any luck:
Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond,
The Schur lectures (1992) (Tel Aviv), 1995, pp. 1–182.MR1321638 (96e:13006)
by Roger Howe.
Does anyone know where I could find a copy?
| https://mathoverflow.net/users/20545 | Reference request: Roger Howe's Schur lectures | This appears to be a (complete?) scanned version:
[Chapters 1-4](http://www.math.ethz.ch/~khorosh/teaching/sym_functions/howe_ch1-4.pdf)
[Chapter 5](http://www.math.ethz.ch/~khorosh/teaching/sym_functions/howe_ch5.pdf)
[Appendix](http://www.math.ethz.ch/~khorosh/teaching/sym_functions/howe_app1-6.pdf)
[Ref... | 14 | https://mathoverflow.net/users/nan | 85608 | 50,952 |
https://mathoverflow.net/questions/85610 | 3 |
>
> **Possible Duplicate:**
>
> [The Klein bottle and the Heawood Conjecture](https://mathoverflow.net/questions/51830/the-klein-bottle-and-the-heawood-conjecture)
>
>
>
It is well known that the [Heawood Conjecture](http://mathworld.wolfram.com/HeawoodConjecture.html) states that the bound for the number of... | https://mathoverflow.net/users/20343 | Klein Bottle exception to the Heawood Conjecture | The derrière cri seems to be [this paper](http://people.math.gatech.edu/~thomas/PAP/5colkb.pdf) (in case the link goes away:
five coloring graphs on the Klein Bottle, by (wait for it):
Chenette, Postle, Streib, Thomas, Yerger.
| 1 | https://mathoverflow.net/users/11142 | 85612 | 50,955 |
https://mathoverflow.net/questions/85593 | 15 | Let $G$ be a semisimple Lie algebra over $\mathbb{C}$. Let $V(\lambda)$ be the irreducible highest weight module for $G$ with highest weight $\lambda$. If $G$ is of type A, we can decompose $V(\lambda)\otimes V(\mu)$ using Littlewood-Richardson rule. In other types, if $\lambda$ and $\mu$ are given explicitly, we can u... | https://mathoverflow.net/users/11877 | Decompose tensor product of type $G_2$ Lie algebras. | The search term you want to look for is "Klimyk's Formula." This formula boils down to the following:
Fix $G$ a compact complex semisimple Lie group. Suppose $V(\lambda)$ and $V(\mu)$ are irreducible representations with highest weights $\lambda$ and $\mu$ respectively. Let $W\_\lambda = \{\lambda\_1,\lambda\_2,\ldot... | 17 | https://mathoverflow.net/users/12301 | 85625 | 50,961 |
https://mathoverflow.net/questions/85563 | 0 | Let $R$ be a semi-ring (resp. a ring).
Let $\hat{R}$ be the structure obtained by "allowing infinite sums" in $R$ so
$$\hat{R} := \lbrace\sum\_{i \in I} a\_i| a\_i \in R\rbrace,$$
where $I$ is countable set.
So my question is:
1. Is $\hat{R}$ a semiring (resp. ring)?
2. If we assume $R \neq 0$ does it follow that $... | https://mathoverflow.net/users/nan | On rings and semirings | It was pointed out in the comments that if we mean by "$\sum\_i a\_i$" just the $I$-tuple $(a\_i: i \in I)$, then the family of such tuples can of course be made into a semi ring (e.g., by pointwise addition and multiplication, or Cauchy multiplication if you use the index set $\mathbb N$). But in this case, neither th... | 2 | https://mathoverflow.net/users/14915 | 85626 | 50,962 |
https://mathoverflow.net/questions/85635 | 15 | **Fermat's Little Theorem**: If $p$ is a prime and $\gcd(a,p)=1$ then $a^{p-1} \equiv1\pmod p$.
Over the years, Fermat's Little Theorem have been generalized in several ways. I am aware of four different generalizations as given below.
**1. Euler**: If $\gcd(a,n)=1$ then $a^{\phi(n)} \equiv1 \pmod n$.
**2. Ramach... | https://mathoverflow.net/users/20174 | Any more generalization of Fermat's Little Theorem? | Fermat's little theorem is a consequence of the fact that the group $(\mathbf{Z}/p\mathbf{Z})^\times$ is cyclic of order $p-1$. Euler's theorem is a consequence of the fact that the (commutative) group $(\mathbf{Z}/n\mathbf{Z})^\times$ has order $\varphi(n)$.
What happens when we replace $\mathbf{Z}$ by the ring of ... | 14 | https://mathoverflow.net/users/2821 | 85637 | 50,968 |
https://mathoverflow.net/questions/85643 | 6 | I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in using nonstandard methods to my research areas, which are in and around arithmetic geometry.
Questions:
1. What kin... | https://mathoverflow.net/users/39742 | Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry | Try this: [www.dpmms.cam.ac.uk/~cb496/nsag1.pdf](http://www.dpmms.cam.ac.uk/~cb496/nsag1.pdf)
and also this <http://wwwmath.uni-muenster.de/u/serpe/documents/ultramath2008serpe-nonstandard-handout.pdf>, [logicandanalysis.org/index.php/jla/article/view/77/29](http://logicandanalysis.org/index.php/jla/article/view/77/29... | 4 | https://mathoverflow.net/users/nan | 85649 | 50,974 |
https://mathoverflow.net/questions/85467 | 21 | Let $X$ be an affine holomorphic symplectic variety of dimension $2n$, with the associated Poisson bracket { , }. Let's say it's an integrable system when there are $n$ algebraically independent holomorphic functions $I\_i$ ($i=1,\ldots,n$) on $X$ such that they Poisson-commute: $\{I\_i,I\_j\}=0$.
Pick a simple Lie a... | https://mathoverflow.net/users/5420 | When is a coadjoint orbit an integrable system (in a weak sense explained below)? | Here is answer (YES) from Alexey Bolsinov who is one the main experts in these questions.
"The answer is YES
There is a very general construction allowing to construct an integrable system on more or less any coadjoint orbit for an arbitrary Lie algebra (non necessarily semisimple). This is a recent paper by Vinber... | 11 | https://mathoverflow.net/users/10446 | 85652 | 50,975 |
https://mathoverflow.net/questions/85651 | 11 | Let $\Gamma$ be one of the classical congruence subgroups $\Gamma\_0(N)$, $\Gamma\_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.
How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?
Any suggestions?
| https://mathoverflow.net/users/10400 | Growth of smallest closed geodesic in congruence subgroups? | For a hyperbolic element $A\in SL(2,\mathbb{Z})$, we have the length of the closed geodesic is given by $\ln[(tr^2(A)-2+\sqrt{tr^4(A)-4tr^2(A)})/4]$, and this is monotonic in $|tr(A)|$ for $|tr(A)|>2$. For $A\in \Gamma(N),\Gamma\_1(N)$, we have $tr(A)\equiv 2 (\mod N)$, and $tr(A)\neq \pm 2$, so the smallest that $|tr(... | 11 | https://mathoverflow.net/users/1345 | 85674 | 50,988 |
https://mathoverflow.net/questions/85493 | 9 | Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach?
Equivalently, we can look at probability distribution on binary sequences of length $a+b$ consisting of $a$ 0's and $b$ 1's, where the probability of ... | https://mathoverflow.net/users/3621 | shape of random q-weighted lattice path | This question is treated as well as the fluctuation problem in a paper I wrote with Dan Beltoft and Cédric Boutillier
<http://arxiv.org/abs/1008.0846>
to appear in Moscow Math Journal.
The approach is based on the use of $q$-Gauss polynomial and the proof is a $q$-analog of the
Moivre Laplace proof of the CLT. Fluctu... | 4 | https://mathoverflow.net/users/20577 | 85680 | 50,993 |
https://mathoverflow.net/questions/85678 | 14 | Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey ["Euler: continued fractions and divergent series (and Nicholas Bernoulli)"](http://www.oswego.edu/Documents/mathematics/EulerDivSeriescfProceedings.pdf), mentions towards the end the continued fraction $$f(x)=... | https://mathoverflow.net/users/29783 | Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant? | The sequence $a\_n=(-1)^n n!$ satisfies $a\_{n+1}+(n+1)a\_n = 0$ (with $a(0)=1$). Thus the generating function satisfies the differential equation $x^2y'+(x+1)y=1$ (where $y(0)=1$). The unique solution is
$$\frac{e^{\frac{1}{x}}Ei\left(1,\frac{1}{x}\right)}{x}$$
For $x=1$, the constant is $e Ei(1,1) \approx .596347362... | 27 | https://mathoverflow.net/users/3993 | 85683 | 50,995 |
https://mathoverflow.net/questions/85560 | 8 | "Examples first:"
Consider so(3,C). (Co)Adjoint Orbits can be described by equations
x^2+y^2+z^2 = R.
R=0 - is nilpotent cone - algebraic closure of the orbit of nilpotent element. (It is union of two orbits - {0} and {Cone w/o {0}} ).
R$\ne$ 0 are orbits of semi-simple elements.
So we have degeneration R->0 - s... | https://mathoverflow.net/users/10446 | Are nilpotent orbits degenerations of semi-simple orbits ? | The answer is often "yes". Here is the sketch of how to obtain a nilpotent orbit as a degeneration of semisimple orbits in the $GL\_n$ case. Let $d$ be a partition of $n$ with $k$ parts and $\overline{\mathcal{O}}\_{d'}$ be the closure of the conjugacy class of nilpotent $n\times n$ matrices with Jordan blocks sizes gi... | 8 | https://mathoverflow.net/users/5740 | 85690 | 51,000 |
https://mathoverflow.net/questions/85693 | 5 | Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi\_k(z)=\sum\_{m>0}a\_{km}z^{-k+m}$). Can one restore a spectral curve that corresponds to this solution?
| https://mathoverflow.net/users/3840 | From Sato grassmannian to spectral curve | This is explained in Segal-Wilson. Essentially, realize your point in the Grassmannian as as a space W of functions w(z), then look for all functions g(z) such that g(z)W is included in W. These functions form a commutative algebra, and Spec of this algebra is your spectral curve. Of course for most points of the Grass... | 8 | https://mathoverflow.net/users/20582 | 85695 | 51,002 |
https://mathoverflow.net/questions/85377 | 12 | It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum\_{j} B\_j \otimes C\_j
$$
with $B\_j$ and $C\_j$ positive semidefinite matrices (of size $n \times n$). For example, it can be seen that the matrix
$$
\begin{pmatrix}
1 & 0 & 0 & 1 \\\
0 & 0 & 0 & 0 \... | https://mathoverflow.net/users/20484 | Decomposition of positive definite matrices. | **The following is just a minor variation of [Martin Argerami's proof of the old question](https://mathoverflow.net/questions/43138/positive-elements-in-tensor-products/43198#43198).** I am even copying his equations and some of his text. If you are +1ing this post, please also +1 his one (if not already done).
Here ... | 11 | https://mathoverflow.net/users/2530 | 85700 | 51,006 |
https://mathoverflow.net/questions/85699 | 7 | There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{\*}$
(by $\omega^{\*}$ I mean set of natural number with reversed order). It seems to be a non-trivial result - for example, one ... | https://mathoverflow.net/users/15129 | Extracting countable chains from linear orders | It seems to me that a natural solution is to use Ramsey's theorem $\aleph\_0 \to (\aleph\_0)^2\_2$: enumerate a countable subset, and color two points depending on whether the enumeration agrees with the given order.
This proof seems "cheaper" to me: Wlog the linear order $P$ is a subset of the rationals. Find a lim... | 9 | https://mathoverflow.net/users/14915 | 85702 | 51,008 |
https://mathoverflow.net/questions/85685 | 4 | Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be integral over $R$; in other words, $R[a]$ is a subring of $K(R)$ that is finite over $R$.
Is it necessarily true that $R... | https://mathoverflow.net/users/5094 | Subtle examples of morphisms that are finite but not flat | Charles, in your [answer](https://mathoverflow.net/a/85692) you're basically discovering the fact that the normalization is not flat (*answer edited to show that it actually **does** provide an answer to the original question*)
Let $X$ be a non-normal reduced scheme and $\sigma:\widetilde X\to X$ its normalization. N... | 19 | https://mathoverflow.net/users/10076 | 85713 | 51,015 |
https://mathoverflow.net/questions/85717 | 6 | Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$.
There are many different categories that one can associate to a space $X$. For example, one could build the small category whose object set is the set of points with only t... | https://mathoverflow.net/users/1622 | Segal's Original Definition of a Topological Category | I would call this an [internal category](http://ncatlab.org/nlab/show/internal+category) in the category of topological spaces and continuous maps.
| 16 | https://mathoverflow.net/users/8508 | 85718 | 51,017 |
https://mathoverflow.net/questions/85716 | 8 | In his [exposé](http://www1.cpm.upmc.fr/videotheque/differe.php?collec=S_C_galois&video=9) at the [Galois bicentenary conference](http://www.galois.ihp.fr/), [Serre](http://www.college-de-france.fr/default/EN/all/historique/essai.htm) makes two references which are not quite explicit.
The first reference occurs (at ... | https://mathoverflow.net/users/2821 | Two implicit references in Serre's *Groupes de Galois : le cas abélien* | I certainly don't claim that I can answer your questions authoritatively , but here are two small remarks.
1) Samuel's book is certainly an excellent guess: it is actually the only textbook in French I can think of entirely devoted to elementary algebraic number theory.
2) There was a preliminary draft of a text ... | 5 | https://mathoverflow.net/users/450 | 85725 | 51,019 |
https://mathoverflow.net/questions/85726 | 2 | Let $G$ be a compact group and $\rho: G \to End(U)$ its linear representation in a finite dimensional vector space $U$. Fix $V \subset U$ - a subspace invariant under $\rho(G)$. Then it is well known that $V$ has some invairant complement.
>
> What are the sufficient conditions (on $G$, $\rho$ or $V$) to
> ensure... | https://mathoverflow.net/users/10847 | Invariant complement to invariant subspace. | This happens if and only if $U$ is expressible as a sum of pairwise non-isomorphic irreducible $G$-modules. If, for example $U \cong V \oplus V$ for an irreducible $G$-module $V,$ then the natural invariant submodule $ \{(v,0): v \in V \}$ has at least two complements: one is the natural choice $\{(0,v): v \in V \}.$ A... | 3 | https://mathoverflow.net/users/14450 | 85729 | 51,020 |
https://mathoverflow.net/questions/85653 | 1 | I am dealing with a vector valued second order homogeneous BVP:
$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$
where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $u$ is a vector valued smooth function on $[0,1]$.
Are there any results that say something about the non-trivial sol... | https://mathoverflow.net/users/17965 | vector valued BVP for ODE's | There are many different ways to obtain sufficient criteria for the nonexistence of nontrivial solutions. Here are some rather simple ways:
1. You can solve the equation explicitly when the coefficients are constant. It is rather straightforward in this situation to identify when nontrivial solutions exist or not.
2.... | 1 | https://mathoverflow.net/users/613 | 85740 | 51,023 |
https://mathoverflow.net/questions/84632 | 2 | $\DeclareMathOperator{\Nat}{Nat}$In a current project, I am trying to "commute" $!$ and $\*$ functors that are both upper or both lower. (Sheaf-theoretic context: constructible étale sheaves.) The fact that they commute when we have one of each ultimately comes down to proper base change: that is, if we have maps $f \c... | https://mathoverflow.net/users/6545 | Equivalent forms of the proper base change isomorphism | (1) is not always an isomorphism when $f$ is an open immersion. (Take $X=Y$ equal to an open subscheme of $Z$, with the obvious maps.) Here is why : when you try to show that the restriction of $g\_\*q\_!$ to the closed complement is $0$, you will want to use a base change isomorphism which is not true in general (it i... | 1 | https://mathoverflow.net/users/12336 | 85748 | 51,027 |
https://mathoverflow.net/questions/85694 | 16 | Metric deformation:
-------------------
Let $(M,g\_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g\_0$, $$g\_t=g\_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric (0,2)-tensor. A natural (and important) question is how the sectional curvatures of $g\_0$ change und... | https://mathoverflow.net/users/15743 | Behavior of sectional curvature under metric deformations | Formula 2) is the correct one in general except it's the derivative of the sectional curvature i.e of $\frac{k\_t(X,Y)}{|X\wedge Y|^2\_t}$ (and not just of $k\_t(X,Y)$) for an orthonormal frame $X,Y$ with respect to the original metric. This accounts for the last term in formula 2.
For $k'(0)$ itself the correct formul... | 16 | https://mathoverflow.net/users/18050 | 85756 | 51,032 |
https://mathoverflow.net/questions/85753 | 5 | Given a braiding $\Psi: X \otimes Y \to Y \otimes X$ for two objects $X,Y$ in a monoidal category, it seems reasonable to assume that $\Psi$ extends uniquely to a braiding $X^k \otimes Y^l \to Y^l \otimes X^k$, for all $k,l \in {\mathbb N}$. The proof would surely be based upon the Yang--Baxter property of $\Psi$ and t... | https://mathoverflow.net/users/11206 | Extending braidings to tensor powers | $\newcommand{\id}{\mathrm{id}} \newcommand{\ot}{\otimes}$
Your assumption is correct. $\Psi$ does extend uniquely to the map that you want. By drawing string diagrams and playing with them, you can intuitively see what to do, namely: every time you see an $X$ strand to the left of a $Y$ strand, use $\Psi$ to braid $X$ ... | 5 | https://mathoverflow.net/users/703 | 85757 | 51,033 |
https://mathoverflow.net/questions/85744 | 7 | Let $X$ be a Noetherian scheme (in particular, we assume that it has only finitely many irreducible components). Is it true that for any open set $U$, the ring $\Gamma(U, \mathscr{O}\_X)$ is a Noetherian ring. Let $\mathscr{F}$ be a coherent sheaf on $X$. Is it true that for any open set $U$, $\Gamma(U,\mathscr{F})$ is... | https://mathoverflow.net/users/11395 | Coherent Sheaves on Noetherian schemes | There exists a noetherian scheme , which is even a variety over a field $k$, such that
$\Gamma(X, \mathscr{O}\_X)$ is not a Noetherian ring.
It is given as Exercise 21.9. D. in Ravi Vakil's wonderful [online book](http://math.stanford.edu/~vakil/216blog/FOAGjan1412public.pdf).
Ravi takes for $X$ the total space... | 10 | https://mathoverflow.net/users/450 | 85762 | 51,035 |
https://mathoverflow.net/questions/85741 | 10 | It's known that the number of representations of an integer $k$ by sum of two squares is
$$
4\;\sum\_{d|k}\left(\frac{-4}{d}\right)
$$
or
$$
4\sum\_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= 4(d\_1(k)-d\_3(k))
$$
where $d\_1(k)$ and $d\_3(k)$ are the numbers of the divisors of $k$ of the forms $4m+1$ or $4m+3$ resp... | https://mathoverflow.net/users/20052 | Representations by positive definite binary quadratic forms | There are going to be very few of these that come out so cleanly. There is not going to be anything clean unless there is only one class per genus "idoneal" (your discriminant $-4N$), and I would not really be confident unless there is only one class entirely for that discriminant. In the latter case, we have your $N=1... | 11 | https://mathoverflow.net/users/3324 | 85763 | 51,036 |
https://mathoverflow.net/questions/85758 | 4 | In Heath-Brown's 2002 paper, "Rational points on curves and surfaces", he states
"We may observe that if $d \geq 3$, the surface $$x\_1^d + x\_2^d - x\_2^{d-2} x\_3 x\_4 = 0$$
is absolutely irreducible, and contains no lines other than those in the planes $x\_2 = 0, x\_3 = 0$, and $x\_4 = 0$."
I am wondering how h... | https://mathoverflow.net/users/10898 | How does one know the following surface contains no other lines? | A line is given by a pair of equations:
\begin{equation\*}
a\_1 x\_1 +a\_2 x\_2+a\_3 x\_3 + a\_4 x\_4=0, \qquad
b\_1 x\_1 + b\_2 x\_2 + b\_3 x\_3 + b\_4 x\_4=0.
\end{equation\*}
Suppose this line is on $X$.
If the minor $a\_3 b\_4-a\_4 b\_3$ is non-zero, then we may rewrite the equations of the line as
\begin{equation... | 12 | https://mathoverflow.net/users/4140 | 85766 | 51,037 |
https://mathoverflow.net/questions/85600 | 16 | The category $\Gamma^{\mathrm{op}}$ is defined to be a skeleton of the category of finite pointed sets (see also [this question](https://mathoverflow.net/questions/74436/geometric-meaning-of-gamma-sets)). Then $\Gamma$-spaces, meaning space-valued presheaves $\Gamma^{\mathrm{op}}\to \mathrm{Spaces}$, can be used to pre... | https://mathoverflow.net/users/49 | What do $\Gamma$-sets classify? | I agree with Charles Rezk's comment. Quite generally, the classifying topos for a universal Horn theory $T$ is the topos of covariant set-valued functors on the category of finitely presented models of $T$. This is proved (twice) in an old joint paper of mine and Andre Scedrov's, "Classifying topoi and finite forcing,"... | 9 | https://mathoverflow.net/users/6794 | 85782 | 51,047 |
https://mathoverflow.net/questions/85793 | 4 | I am looking for the original proof by Borsuk of the Borsuk-Ulam theorem. I would appreciate very much if someone could outline the proof.
| https://mathoverflow.net/users/20620 | Original proof of the Borsuk-Ulam theorem | As J.C. Ottem suggests, the best thing in those cases is to look for the original paper. [Wikipedia](https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem) says
>
> According to (Matoušek 2003, p. 25), the first historical mention of the statement of this theorem appears in (Lyusternik 1930). The first proof wa... | 12 | https://mathoverflow.net/users/7460 | 85794 | 51,054 |
https://mathoverflow.net/questions/85780 | 5 | I would like to ask a couple of naive question about the following theorem of Max Noether:
<http://en.wikipedia.org/wiki/AF%2BBG_theorem>
In the book of Fulton, page 60
<http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf>
this theorem is called *Max Noether's fundamental theorem*.
**Question 1.** Are ther... | https://mathoverflow.net/users/13441 | Max Noether's residual intersection theorem (Fundamentalsatz): importance and applications | Perhaps one of most famous consequences of Noether "AF+BG Theorem" is Cayley-Bacharach Theorem, that I state below.
>
> **Theorem (Cayley-Bacharach)**. Let $X\_1, X\_2 \subset \mathbf{P}^2$ be two plane curves of degree $d$ and $e$, respectively, meeting in a collection of $d \cdot e$ distinct points $\Gamma$. If $... | 7 | https://mathoverflow.net/users/7460 | 85799 | 51,056 |
https://mathoverflow.net/questions/85802 | 1 | By means of the Hopf fibration $\pi: S^3 \rightarrow S^2$, U. Pinkall showed that every compact Riemannian surface of genus one can be conformally embedded in $S^3$.
More precisely:
Let $p$ be a closed curve on $S^2$ of length $L$. Lifting it to $S^3$ yields a torus isometric to $R^2 / \Gamma$, with $\Gamma$ generate... | https://mathoverflow.net/users/18589 | Hopf Tori in $S^3$ | Your definition of the Clifford torus is off. The usual definition of the Clifford torus is the set $(z\_1,z\_2)\in\mathbb C^2$ in the unit sphere $|z\_1|^2+|z\_2|^2=1$ with $|z\_1|^2=|z\_2|^2=\frac 1 2$. This is a square torus isometric to $\mathbb R^2/\Gamma\_c$ with $\Gamma\_c$ generated by $(2\pi/\sqrt 2, 0), (0, 2... | 5 | https://mathoverflow.net/users/18050 | 85811 | 51,062 |
https://mathoverflow.net/questions/85791 | 13 | I have a question about the proof of Theorem 5 in Thurston's paper "A norm for the homology of 3-manifolds". It's the theorem that asserts that the fibered faces are, well, fibered. More precisely, fix a compact 3-manifold $M$, and let $B\_x$ be the Thurston polytope. Then the theorem asserts that the set of elements o... | https://mathoverflow.net/users/20619 | Question about Thurston's paper "A norm for the homology of 3-manifolds" | It's been a very long time since I've read this paper, and I haven't been able to find a copy online, so my apologies in advance if what I'm about to write is nonsense.
Think in terms of projective coordinates (i.e. I won't say "cone" any more). I think the lemma you describe also holds for surfaces $S$ on the bounda... | 8 | https://mathoverflow.net/users/284 | 85814 | 51,064 |
https://mathoverflow.net/questions/85813 | 0 | Why does on a scheme $X$ of characteristic zero for a vector bundle $F$ (of finite rank) the operation $Sym^n(.)$ commute with taking the dual bundle $(.)^v$, i.e.
$Sym^n(F^v) \simeq Sym^n(F)^v$ canonically ?
How does the iso work explicitly?
| https://mathoverflow.net/users/18183 | Vector bundles and tensor operations | There is always a map $Sym^{n}(F^{\vee}) \rightarrow Sym^{n}(F)^{\vee}$, regardless of the characteristic, and this map is an isomorphism in characteristic $0$. The map is induced
by the natural pairing $Sym^{n}(F^{\vee}) \times Sym^{n}(F) \rightarrow \mathcal{O}\_{X}$, given locally by $(\phi\_{1}\cdots \phi\_{n}, f\_... | 2 | https://mathoverflow.net/users/4659 | 85818 | 51,066 |
https://mathoverflow.net/questions/85835 | 8 | Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask about the following specific way to describe the distribution in terms of the variance of tails.
Given a number t betwee... | https://mathoverflow.net/users/1532 | A Variance-Tail Description for Continuous Probability Distributions | For a normal distribution with mean $0$ and standard deviation $\sigma$, I get (with help from Maple)
$$W\_X(t) = \sigma^2 + \sqrt{\frac{2}{\pi}} \frac{\sigma t}{1 - \text{erf}(t/(\sqrt{2}\sigma))}
e^{-t^2/(2 \sigma^2)} + \frac{2 \sigma^2}{\pi (1 - \text{erf}(t/(\sqrt{2}\sigma))^2}
e^{-t^2/\sigma^2}$$
As $t \to \inft... | 6 | https://mathoverflow.net/users/13650 | 85840 | 51,075 |
https://mathoverflow.net/questions/85836 | 2 | Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x\_1, \ldots , x\_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[x\_1, \ldots , x\_n]$? (We can suppose that every ideal in $R$ is principal)
| https://mathoverflow.net/users/20272 | Number of generators of an ideal in a polynomial ring over a Noetherian ring | Nothing. Assume $R=k$, a field, for specificity. Then $k[x\_1]$ is a principal ideal domain, as you know, but $k[x\_1,x\_2]$ has ideals with unbounded number of generators. Specifically, $(x\_1,x\_2)^n$ is minimally generated by $n+1$ elements for all $n$. One can get higher rates of growth by adding more variables.
| 14 | https://mathoverflow.net/users/460 | 85849 | 51,078 |
https://mathoverflow.net/questions/85787 | 13 | I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I alluded to this in a previous question
[Fitting a mesh to a density function](https://mathoverflow.net/questions/85624... | https://mathoverflow.net/users/11828 | 2/3 power law in the plane | It is a theorem of Renyi and Soulanke that the cardinality of the boundary of a convex hull of a uniformly distributed random point set of cardinality $N$ in a smooth convex set grows like $N^{1/3},$ so in particular, if you take a point set in a disk of radius $R,$ so that the density is $1,$ then the cardinality of s... | 3 | https://mathoverflow.net/users/11142 | 85853 | 51,081 |
https://mathoverflow.net/questions/85850 | 6 | The Kauffman bracket skein module $K\_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). It is a quantum deformation of the coordinate ring of the character variety $\operatorname{Hom}(\pi\_1(F),\operatorname{S... | https://mathoverflow.net/users/35353 | What vector space does the Kauffman bracket skein algebra of FxI act on? | The answer is $K\_t(H)$, where $H$ is a handlebody with boundary $F$. If $t$ is a root of 1 and we are taking the usual semisimple quotient, then $K\_t(F\times I)$ is isomorphic to a matrix algebra and $K\_t(H)$ isomorphic to the standard representation. Also, in this case we can let $H$ be *any* 3-manifold with bounda... | 6 | https://mathoverflow.net/users/284 | 85864 | 51,087 |
https://mathoverflow.net/questions/85856 | 8 | Here is what I know about Landau-Ginzburg models:
It is an important player in the story of mirror symmetry.
It involves "potentials" which are functions of complex varibles, which have isolated singularities at the origin.
There is the notion of universal unfolding(miniversal unfolding, or miniversal deformation) at... | https://mathoverflow.net/users/4624 | Places to learn about Landau-Ginzburg models | I think the Clay Math books have nice descriptions about Landau-Ginzburg Models. The book called [Mirror Symmetry](http://www.claymath.org/publications/Mirror_Symmetry/) and [Dirichlet Branes and Mirror Symmetry](http://www.claymath.org/publications/Dirichlet_Branes/) both have nice physical and mathematical descriptio... | 5 | https://mathoverflow.net/users/2565 | 85866 | 51,088 |
https://mathoverflow.net/questions/85730 | 11 | Suppose we have some positive quantites $P$ and $Q$ which depend on some choices that we make, and we want to show that some choice makes the quotient $P/Q$ fall below some cool bound.
One idea is to make our choices randomly in some way and show that $P/Q$ is small on average. That is, we can use the trivial bound
$... | https://mathoverflow.net/users/20598 | A trick or a general technique? (Probabilistic Method) | One way to see this technique is as a way of dealing with certain bad cases. $E[P/Q]$ can be unhelpfully dragged up by the inclusion of certain cases where $Q$ is small and $P$ is medium. $E[P]/E[Q]$ is not nearly so distorted. In particular, take $Q=0$, $P>0$. The first inequality becomes totally unhelpful, as $E[P/Q]... | 11 | https://mathoverflow.net/users/18060 | 85877 | 51,094 |
https://mathoverflow.net/questions/85873 | 5 | In some calculations I am writing up,
$\newcommand{\cR}{{\mathcal R}}$
I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be embedded inside $\cR$, the hyperfinite ${\rm II}\_1$. I think I have managed to hack out an explicit embedding, by real... | https://mathoverflow.net/users/763 | Reference for embedding an infinite direct product of matrix algebras into the hyperfinite $II_1$ factor | Murray and von Neumann showed that if $p \in \mathcal R$ is a non-zero projection then $p\mathcal R p \cong \mathcal R$, i.e., the fundamental group of $\mathcal R$ is all positive reals (You should be able to find this in most books that discuss the hyperfinite II$\_1$ factor. Also, note that this is easy to see if $p... | 11 | https://mathoverflow.net/users/6460 | 85878 | 51,095 |
https://mathoverflow.net/questions/85844 | 6 | Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is *geodesically convex* if for any $x,y\in E$ the **shortest** geodesic connecting $x$ and $y$ lies in $E$.
**Question.** How large can the Haar/Lebesgue measure of $E$ can be?
For example, is $d=2$, then it seems t... | https://mathoverflow.net/users/8131 | Large geodesically convex subsets of tori | *(This is a new answer; my original answer was completely wrong.)*
Assume $\mathop{\rm vol}E>\tfrac12$.
Then it contains two opposite points say $x$ and $x'=x+(\tfrac12,\tfrac12,\dots,\tfrac12)$.
WLOG we can assume that $x=0$.
Taking minimizing geodesics form $(\tfrac12,\tfrac12,\dots,\tfrac12)$ to $y\approx 0$, we g... | 4 | https://mathoverflow.net/users/1441 | 85890 | 51,101 |
https://mathoverflow.net/questions/85842 | 8 | Prove that doesn't exist $N\in\mathbb{N}$ with property: for all primes $p>N$ exist $n\in\{3, 4,\ldots, N\}$ such that $n, n-1, n-2$ are quadratic residues modulo $p$.
| https://mathoverflow.net/users/20643 | Three consecutive quadratic residues problem | By Dirichlet's theorem, there exists $p>N$ such that each prime $l\leq N$,
with the exception of $l=3$, satisfies $(l/p) = (l/3)$. I claim that
this $p$ is a counterexample. Indeed by multiplicativity $(m/p) = (m/3)$
for each $m \leq N$ that is not a multiple of 3. In particular $(m/p) = -1$
if $m \equiv -1 \bmod 3$. E... | 20 | https://mathoverflow.net/users/14830 | 85891 | 51,102 |
https://mathoverflow.net/questions/85868 | 1 | Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$.
For every continuous linear functional $F$ on $B$, define
$V(F)=min\_{c\epsilon C} F(c)$ and
$S(F)= { \lbrace c \epsilon C : V(c)=V(F)\rbrace }$.
Is it true (or are there known conditions under which)
$V:B^\*\rightarrow \mathbb{R}$ is Frech... | https://mathoverflow.net/users/20211 | Conditions for differentiability of minima and minimizers of linear functionals? | In general $V$ is Lipschitz, but it is not smooth: take e.g. $B=\mathbb{R^2}$ with the Euclidean norm, and let $C$ be a segment, say $C:=[-1,1]\times(0)$. Then, for all $(x,y)\in B^\*=B$ we have $$V(x,y)=\min\_{|c|\le1} \ cx = \min \{x,-x\}\, .$$ To get a smooth $V$, smoothness conditions on $C$ are required.
| 3 | https://mathoverflow.net/users/6101 | 85893 | 51,104 |
https://mathoverflow.net/questions/85894 | -1 | If $G$ is a finite group, and $n$ is the least positive integer such that $G$ can be embedded in symmetric group $S\_n$, then, should $G$ necessarily contain a subgroup of index $n$?
| https://mathoverflow.net/users/6761 | Embedding of finite groups in Symmetric Groups | This is not true. The group $S\_3 \times S\_2$ can be embedded in $S\_5$, but not in smaller symmetric groups.
| 5 | https://mathoverflow.net/users/17036 | 85895 | 51,105 |
https://mathoverflow.net/questions/85720 | 1 | I am considering the following problem:
(i) Fix $n$ and color the edges of $K\_n$ red and blue arbitrarily.
(ii) Let $M$ be the set of monochromatic triangles in $K\_n$ and define $g:M\rightarrow \mathbb{N}$ as $g(T\_{xyz})$= $|N\_r(x)\cap N\_r(y)\cap N\_r(z)|$ if $T\_{xyz}$ is red and $|N\_b(x)\cap N\_b(y)\cap N\... | https://mathoverflow.net/users/14875 | Combinatorial optimization and graph coloring | Linear lower bound. For any 18 vertices we have a monochromatic quadrilateral by Ramsey theorem. It follows that the total number of monochromatic quadrilaterals is not less then ${n\choose 18}/{n-4\choose 14}$. Then one of triangles is contained in at least at least $4{n\choose 18}/({n-4\choose 14}\cdot {n\choose 3})=... | 4 | https://mathoverflow.net/users/4312 | 85897 | 51,107 |
https://mathoverflow.net/questions/85881 | 19 | I'm betting `yes, sure!', but don't see it. Could someone please point me toward,
or construct for me, a Lagrangian submanifold immersed in
standard symplectic ${\mathbb R}^{2n}$ for $n > 1$, whose closure is all of ${\mathbb R}^{2n}$?
(For an $n =1$ example, one can use the leaves arising from
[this modification ... | https://mathoverflow.net/users/2906 | Can a Lagrangian submanifold of ${\mathbb R}^{2n}$ be dense ($n>1$)? | Your question already has the answer in it for $n=2$. Take a connected complex curve $L\subset\mathbb{C}^2$ that is dense in $\mathbb{C}^2$. Then $L$ is Lagrangian for the real part of the holomorphic $2$-form $\Upsilon = dz^1\wedge dz^2$. This real part of $\Upsilon$ is equivalent to the standard symplectic structure ... | 22 | https://mathoverflow.net/users/13972 | 85901 | 51,109 |
https://mathoverflow.net/questions/85903 | 2 | Hello,
I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:
>
> Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. Let $M$ be a f.g. $A$-module. Then $Ext^i (M,A)=0$ for $i < codim(supp(M))$ (codimension of support).
>
>
>
The... | https://mathoverflow.net/users/2095 | Modules with small support have big depth - reference wanted | For $A$ local, your statement follows from a result of Ischebeck. What you want holds even if $A$ is just Cohen-Macaulay. More generally $\mathrm{Ext}^i\_A(M,N)=0$ for $i<\mathrm{depth}\ N-\dim M$. Put $N=A$ and assume $A$ is Cohen-Macaulay, then you get $\mathrm{Ext}^i\_A(M,A)=0$ for $i<\dim A-\dim M$.
You can find ... | 3 | https://mathoverflow.net/users/16046 | 85905 | 51,111 |
https://mathoverflow.net/questions/85871 | 0 | I speculated in 2008 that the modified Neretin polynomials presented in [A145900](http://oeis.org/A145900) of the On-line Encyclopedia of Integer Sequences, which can be summed to give a normalized Schwarzian derivative for a complex function and are related to a representation of the Virasoro algebra, all have integer... | https://mathoverflow.net/users/12178 | Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients | Before I give the answer, let me try to formulate the question in the way *I* would have asked it here. Fortunately I am not bound by the OEIS requirements of brevity and ASCII, and there is LaTeX here...
**Question.** Let $A$ be the polynomial ring $\mathbb Z\left[c\_1,c\_2,c\_3,...\right]$ in infinitely many commut... | 7 | https://mathoverflow.net/users/2530 | 85911 | 51,115 |
https://mathoverflow.net/questions/85915 | 8 | Let $\mathbf{R}$ be the field of real numbers. What are the generators of the maximal ideals of the polynomial ring $\mathbf{R}[x\_1, ... , x\_n]$? If instead of $\mathbf{R}$ one considers the field $\mathbf{C}$ of complex numbers, then Hilbert's Nullstellensatz implies that each maximal ideal $\mathfrak{m}$ of $\mathb... | https://mathoverflow.net/users/20272 | Maximal ideals in a polynomial ring over the real numbers. | There are two kind of maximal ideals in $\mathbf{R}[x\_1, \ldots, x\_n]$: the ideals corresponding to real points of $\mathbf{A}^n\_{\mathbb{R}}$, i.e. of the form $$(x\_1-a\_1, \ldots, x\_n-a\_n), \quad a\_i \in \mathbf{R}$$
and the ideals corresponding to pairs of complex-conjugated points, that after a real change o... | 21 | https://mathoverflow.net/users/7460 | 85916 | 51,118 |
https://mathoverflow.net/questions/85908 | 7 | Let $X$ be a smooth projective variety over $\mathbb{C}$, and fix $A$ an ample divisor as the polarization. We say a vector bundle $E$ to be (semi-)stable, if for any proper subsheaf $F$ of $E$, $\mu(F)<\mu(E)$ (resp. $\leq$).
I guess it is not sufficient to check just subbundles $F$ of $E$. But is there a counterex... | https://mathoverflow.net/users/4975 | Stability condition for vector bundles | There are many examples of unstable bundles on a projective surface that have no non-trivial subbundles. For example, if $k$ is an integer with $k < 3$ and $I$ is the sheaf of ideal of $m$ distinct points in $\mathbb P^2$, with $m > 0$, there exists an extension
$$
0 \longrightarrow \mathcal O \longrightarrow E \longri... | 9 | https://mathoverflow.net/users/4790 | 85922 | 51,121 |
https://mathoverflow.net/questions/85920 | 4 | Hi Everyone,
I am a math amateur who for the past year has been working on better understanding Bertrand's Postulate, the Ramanujan Primes, and the recent expansion of Bertrand's Postulate (always a prime between 2x and 3x and always a prime between 3x and 4x) using elementary methods.
I've been working with least ... | https://mathoverflow.net/users/15915 | Least Prime Factors: found a counting formula for a given range -- what is the standard approach? | First I'll toot my own horn.
There is still some work left for elementary and near elementary methods to accomplish. Based on your description, I think your formulas say something about the distribution of numbers coprime to the kth primorial. I have been working on something similar, and part of the path has led me ... | 4 | https://mathoverflow.net/users/3402 | 85924 | 51,122 |
https://mathoverflow.net/questions/85311 | 2 | I have a polynomial:
$$f(x\_1 \dots x\_n) = \prod\_{i=1}^n (c\_ix\_i + 1) - \frac{1}{2}c\_0\sum\_{i=1}^nx\_i^2$$
Given some values for $c\_0 \dots c\_n$, I'd like to choose the maximizing values for $x\_1 \dots x\_n$. I'm not concerned with the actual maximum value of $f(x\_1 \dots x\_n)$, only the values of the in... | https://mathoverflow.net/users/20470 | Optimization of a Specific Polynomial | As noted by Pietro in the comments only a few combinations of $n$ and coefficients will even *have* a finite maximum, unless you bound your domain. For example, consider $c\_0=-2$ and all other $c\_i=0$. Even this is unbounded above.
Now, let's assume you either bound your domain, or have ensured that the polynomial ... | 2 | https://mathoverflow.net/users/20665 | 85930 | 51,124 |
https://mathoverflow.net/questions/85935 | 2 | Consider the canonical symplectic structure $(\omega, J)$ on $\mathbb{R}^{2n}$.
(i) What can be said about the density of the lagrangian grassmannian $L$ (i.e. those rank $n$ totally isotropic linear subspaces of $\mathbb{R}^{2n}$) in the usual Grassmannian $Gr=Gr\_{n}(\mathbb{R}^{2n})$ of rank $n$ linear subspaces? ... | https://mathoverflow.net/users/20516 | density of lagrangian grassmannian in usual grassmannian. | jmart has modified the question since I posted my original answer. I'll now modify my answer to correspond:
(i) The Lagrangian Grassmannian ($L$ in your notation) is a closed submanifold of $Gr\_n(\mathbb{R}^{2n})$, so it's not dense at all. In fact, it has dimension $\frac12n(n{+}1)$.
(ii) $L$ is homogeneous unde... | 7 | https://mathoverflow.net/users/13972 | 85939 | 51,129 |
https://mathoverflow.net/questions/85946 | 1 | How would you explain [one of these theorems in the foundations of mathematics](http://en.wikipedia.org/wiki/Category%3ATheorems_in_the_foundations_of_mathematics) to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the details?
The aforementioned link ... | https://mathoverflow.net/users/20215 | An undergraduate's guide to the foundational theorems of logic | **Edit:** This answer was given to the original formulation of the question, which asked for five-minute explanations for laypersons met on the street, rather than handwavy introductions for undergraduates. Maybe it still works though.
---
Since I have only 5 minutes to tell a layperson, I'd channel the late Geor... | 8 | https://mathoverflow.net/users/4137 | 85951 | 51,132 |
https://mathoverflow.net/questions/85961 | 8 | Where can I find a book which explains the development of modern logic, e.g. Tarski, Frege, Peano, up untill Wittgenstein, Russel?
| https://mathoverflow.net/users/20696 | History of Logic Development | The scope of the figures you mention (Tarski, Frege, Peano, Wittgenstein, Russell) makes it a little unclear exactly what you're after. For instance, *From Frege to Goedel* (as mentioned by Mahmud) is an excellent compilation of early texts in mathematical logic -- you get e.g. Frege, Peano, Hilbert, Zermelo, Skolem, H... | 7 | https://mathoverflow.net/users/4137 | 85965 | 51,138 |
https://mathoverflow.net/questions/85956 | 1 | The following is a theorem of which I have great interest in but cannot find anything about on the internet,
>
> Every 3-manifold of finite volume comes from identifying sides of some polyhedron
>
>
>
I'm fairly certain that "identifying sides of some polyhedron" may be a simplification of the technical termin... | https://mathoverflow.net/users/20343 | 3-manifold theorem reference request or proof | I think the reference that you are looking for is [this article](http://arxiv.org/pdf/0806.1912) by Cannon, Floyd, and Parry.
| 6 | https://mathoverflow.net/users/36108 | 85971 | 51,142 |
https://mathoverflow.net/questions/85941 | 7 | Let $\text{ZF}^-$ be the set theory without powerset, choice, and foundation. Consider the following notions:
* Wellfounded sets
$$WF(c) \Leftrightarrow (\forall x \subseteq TC(c)) \left[x \neq \emptyset \rightarrow (\exists y \in x) (\forall t \in x) [t \notin y]\right]$$
* Ordinals
$$ON(c) \Leftrightarrow WF(c) \we... | https://mathoverflow.net/users/20684 | Is $\omega$ absolute in set theory without foundation? | None of these three notions is absolute, even if you retain powerset and the axiom of choice.
In Boffa’s set theory (which contains ZFC without foundation, and is conservative over ZFC with respect to the well-founded kernel), every extensional set-like binary relation is isomorphic to a transitive class with $\in$. ... | 11 | https://mathoverflow.net/users/12705 | 85981 | 51,146 |
https://mathoverflow.net/questions/85985 | 10 | From many sources I am hearing that symmetric polynomials not only an algebra, but also Hopf algebra.
Would someone be so kind to explain where the coproduct (antipode) comes from ?
And what it is useful for ?
It is mentioned e.g. here:
[Avatars of the ring of symmetric polynomials](https://mathoverflow.net/question... | https://mathoverflow.net/users/10446 | Symmetric polynoms are Hopf algebra ? What for one needs co-product ? | Yes, there exists a natural Hopf algebra structure on the ring of symmetric functions (i. e., symmetric "polynomials" in infinitely many indeterminates). It is not related to the additive group of $k^n$ (for a good reason: as you said, the obvious coalgebra structure on $k\left[x\_1,x\_2,...,x\_n\right]$ coming from ad... | 21 | https://mathoverflow.net/users/2530 | 85990 | 51,151 |
https://mathoverflow.net/questions/85329 | 10 | Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following *unit axiom* not considered in other references (e.g. Schwede-Shipley): given a cofibrant replacement of the monoidal unit $q\colon QI\stackrel{\sim}\rightarrow I$ and a cofibrant object $X$, the morphisms $q\otimes X$... | https://mathoverflow.net/users/12166 | Hovey's unit axiom in monoidal model categories | I can only answer half of your question: namely, a standard condition under which the more general unit axiom holds. I don't know of any examples where Hovey's unit axiom holds but this more general one does not. The hypothesis is that **cofibrant objects are flat**, i.e. smashing with cofibrant objects preserves weak ... | 5 | https://mathoverflow.net/users/11540 | 85995 | 51,154 |
https://mathoverflow.net/questions/85927 | 5 | I asked this question earlier on [math.stackexchange.com](https://math.stackexchange.com/questions/99015/counting-ordered-tuples-with-an-additional-condition) but didn't get an answer:
>
> Let $0 < a\_1 < \cdots < a\_n$ be integers. Is there a closed formula (or some other result) for the number $N(a\_1,\ldots,a\_n... | https://mathoverflow.net/users/10194 | Number of integer combinations $x_1 < \cdots < x_n$? | Robin Pemantle and Herb Wilf give a short recurrence as an answer to this question, and a more compact formula when the sequence $a\_n$ is linear, in a freely available paper from the EJC in 2009: vol. 16 (2009), #R60, "Counting Nondecreasing Integer Sequences that Lie Below a Barrier." Link: <http://www.combinatorics.... | 7 | https://mathoverflow.net/users/12878 | 85997 | 51,155 |
https://mathoverflow.net/questions/86000 | 10 | Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify this subset with $X$) and such that $\overline X\setminus X$ is a proper analytic subset (with no conditions on its codi... | https://mathoverflow.net/users/9871 | Non-bimeromorphic compactifications | As you guessed, the answer is **no**.
The following counterexample in dimension $2$ can be found in Vo Van Tan's paper [On the compactification of strongly pseudoconvex surfaces III](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0195&DMDID=dmdlog32), *Mathematische Zeitschrift* **195** (1987), 259-... | 10 | https://mathoverflow.net/users/7460 | 86004 | 51,157 |
https://mathoverflow.net/questions/85994 | 3 | Let $J$ be the $n$ by $n$ matrix whose each entry is $1$. Also define $f(n)$ to be the least $m$ so that there is a $\lambda>0$ so that $\lambda J$ is the sum of at most $m$ unitary matrices. Note $f(2)=2$.
What is the value of $f(n)$ for $n>2$?
| https://mathoverflow.net/users/14885 | Sums of Unitary Matrices | Edit: I correct the mistake pointed out by Matthew in his comment.
In fact, any matrix $X$ can be written as $X=\lambda(U+V)$ for unitary matrices $U,V$, and $\lambda$ can be taken as a half of the operator norm of $X$. For a proof, see [this](https://mathoverflow.net/questions/50936) question. This is slightly stron... | 7 | https://mathoverflow.net/users/10265 | 86010 | 51,163 |
https://mathoverflow.net/questions/85976 | 2 | It seems easy but I can't prove it. Can anyone give proof or reference?
| https://mathoverflow.net/users/6569 | Why Tristram-Levine signature jumps at the zeros of alexander polynomial? | If $A$ is a Seifert matrix for $K$ and $\omega \in \mathbb{C}$ has norm 1, then the Tristram-Levine signature $\sigma\_\omega(K)$ is the signature of the matrix
$(1-\omega)A + (1-\bar{\omega})A^T = (1-\bar{\omega})(A^T - \omega A),$
which jumps when some eigenvalue of $A^T - \omega A$ crosses zero (i.e. changes si... | 4 | https://mathoverflow.net/users/428 | 86014 | 51,167 |
https://mathoverflow.net/questions/86019 | 8 | Hi,
I am looking for a software package that will allow me to experiment with the irreducible representations of lie groups (for example, $SL(2,p)$) over the complex field and over finite fields. That is, I would like to get the corresponding matrices for group elements.
Thanks,
Shachar
| https://mathoverflow.net/users/20709 | Software package to manipulate representations | For representations over the complex field, I know that GAP does a good job. (I'm not sure if it can do modular representations as well, but I wouldn't be surprised.)
Here is some example code to get you started:
```
G:=SL(2,3);;
reps:=IrreducibleRepresentations(G);;
Elements(G);
List(G,g->g^reps[5]);
```
This p... | 10 | https://mathoverflow.net/users/9068 | 86023 | 51,170 |
https://mathoverflow.net/questions/86016 | -1 | Using the formalism of model categories its possible define the concept of homotopy as done [here](http://ncatlab.org/nlab/show/homotopy).
If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and fibration and co-fibration defined in the standard topological way, these type of h... | https://mathoverflow.net/users/14969 | Alternative characterization of homotopy equivalence | EDIT: Now that the OP has edited his question to make clearer what he wants as an answer, I'm removing speculation about what he wanted. The answer is: yes, you can characterize homotopy equivalences as the maps which become isomorphisms after applying the localization functor to invert the weak equivalences. This answ... | 3 | https://mathoverflow.net/users/11540 | 86032 | 51,173 |
https://mathoverflow.net/questions/86033 | 4 | **Definition:** Let $R$ be an $A$-algebra, where $R$ and $A$ are both commutative rings with unit. Let $S \subset R$ be the (multiplicatively closed) subset consisting of those nonzerodivisors $s \in R$ such that $R/sR$ is flat over $A$. I will call $R$ *integrally closed over $A$* if it satisfies
1. $R$ is flat over... | https://mathoverflow.net/users/5094 | Relative integral closure | $S$ consists of polynomials whose coefficients are not in any ideal, that is, polynomials whose coefficients generate the unit ideal.
First assume $A$ is integral. Let $K$ be the field of fractions of $A$, then clearly $K[x\_1,...,x\_n]$ is an integrally closed unique factorization domain. Given an element in this th... | 4 | https://mathoverflow.net/users/18060 | 86038 | 51,177 |
https://mathoverflow.net/questions/83906 | 7 | Suppose $X$ is a DM stack, and let $E^\bullet$ be a perfect obstruction theory of $X$ such that the $E^{-1}$ term admits a trivial quotient/sub-bundle. Is it true that the virtual fundamental class $[X, E^\bullet]$ is zero?
If $X$ is smooth, then this is true: In such a case, the virtual fundamental class is the top ... | https://mathoverflow.net/users/1703 | Trivial obstructions and virtual fundamental classes | It turns out that this is true. In the paper "Localizing Virtual Cycles by Cosections" by Kiem and Li, they address the case where one has a surjection $Ob \to \mathcal{O}$. In the case of an injection $\mathcal{O} \to Ob$, one can produce via a diagram chase a corresponding surjection, which yields the claim.
| 4 | https://mathoverflow.net/users/1703 | 86057 | 51,187 |
https://mathoverflow.net/questions/86058 | 2 | Is it possible to have a saturated ideal on a successor cardinal which does not extend the nonstationary ideal? (i.e. some nonstationary set is positive for this ideal)
| https://mathoverflow.net/users/11145 | saturated ideals | Yes, it is. The reason is that an ideal $I$ on $P(\kappa)$ is saturated just in case the quotient Boolean algebra $P(\kappa)/I$ satisfies the $\kappa^+$-chain condition, and this is a property that is preserved by permutations of the underlying set $\kappa$. But the property of extending the non-stationary ideal is not... | 4 | https://mathoverflow.net/users/1946 | 86059 | 51,188 |
https://mathoverflow.net/questions/86049 | 1 | Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\forall x$.
The easiest examples of such functions are constant functions.
Dirichlet's function ($1$ if $x\in\mathbb Q$ and $0$... | https://mathoverflow.net/users/20692 | Is there a periodic function without minimum period such that all the possible periods are irrationals? | I guess what you can get as a set of periods is exactly any additive subgroup of the reals. Certainly the periods are closed under addition. On the other hand, for any subgroup $G$ or $\mathbb R$, mimic the Dirichlet function by defining $f$ to be the indicator function of $G$. Here the set of periods is exactly $G$ it... | 7 | https://mathoverflow.net/users/11054 | 86060 | 51,189 |
https://mathoverflow.net/questions/86066 | 9 | I would like to compute the Tutte polynomial of the complete graph $K\_n$ for n as large as possible. Using a program by Björklund, Husfeldt, Kaski, Koivisto ([here](https://github.com/thorehusfeldt/tutte_bhkk)), I managed to compute up to n=18 on my home computer (in serial) in less than a day. Overall, I've been very... | https://mathoverflow.net/users/2264 | How many Tutte polynomials of complete graphs are known? | Our program "tutte" (<http://homepages.ecs.vuw.ac.nz/~djp/tutte/#download>) can compute the TP of $K\_{18}$ in 160s (on a recent machine with an i7).
However you wouldn't want to do it with this sort of program that works on general graphs, because the complete graph is special and by tackling it symbolically you can... | 8 | https://mathoverflow.net/users/1492 | 86070 | 51,195 |
https://mathoverflow.net/questions/86064 | 9 | I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for the following reason: I always tell them that they are one of the most important topics for applications of calculus (thi... | https://mathoverflow.net/users/4358 | Differential Equation Examples for Calculus Students | This [worksheet](http://www.math.washington.edu/~m125/Worksheets/DiffEQ.pdf) from Dept. of Mathematics in University of Washington guides through two examples using differential equations: forensic mathematics and spread of rumor.
Also the internet magazine called +plus magazine has [many examples of applications](ht... | 7 | https://mathoverflow.net/users/20215 | 86072 | 51,197 |
https://mathoverflow.net/questions/86048 | 6 | In Chapter 6 of Mumford's Geometric invariant theory, during the proof of the rigidity lemma, there are two statements I'm not sure how to verify. The general setup is:
$p : X \rightarrow S$ is flat, $S$ connected, and $H^0(X\_s, o\_{X\_s}) \cong k(s)$ for all points $s \in S$.
1. In the first part, we're assuming ... | https://mathoverflow.net/users/18403 | Verifying claims in the proof of the Rigidity Lemma (Mumford, GIT) | 1) This is really simple. If $S=\{s\}$, then $X=X\_s$ and hence $p\_\*\mathscr O\_X=H^0(X,\mathscr O\_X)=k(s)=\mathscr O\_S$.
2) This may be a little trickier, but still not too hard.
-- Since the statement is local on $S$, we may assume that $S=\mathrm{Spec}A$ is affine.
-- We may also assume that $X=\mathrm{... | 4 | https://mathoverflow.net/users/10076 | 86074 | 51,198 |
https://mathoverflow.net/questions/86047 | 1 | **Background:**
Let $\mathbb{F}$ be an algebraically closed field. Let $X \subset \mathbb{F}^n$ be an affine variety. Let $\pi(X)$ be the projection of $X$ to the first $m < n$ coordinates,
$$
\pi(X) = \{(x\_1,\ldots,x\_m): x \in X\},
$$
and for a point $a \in \mathbb{F}^m$ let $\phi(a,X)$ be the fiber of $X$ over $... | https://mathoverflow.net/users/20709 | Degree of fibers with too large dimension | This is an interesting idea, but I think there are some issues with it. First of all, it seems to me that if the image $\pi(X)$ has degree $>1$, then all fibers will have strictly smaller degree than $\mathrm{deg}X$. So one might ask, if it is true that the fibers with excess dimension have smaller degree than those fi... | 2 | https://mathoverflow.net/users/10076 | 86077 | 51,201 |
https://mathoverflow.net/questions/86080 | 4 | One of the requirements for a smooth manifold $M$ is that it be [paracompact](http://en.wikipedia.org/wiki/Paracompact_space), and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there exists a smooth [partition of unity](http://en.wikipedia.org/wiki/Partitio... | https://mathoverflow.net/users/2051 | Is the space of smooth partitions of unity connected? Simply-connected? | Typically, partitions of unity are used to prove a statement along the following lines. Given a paracompact $X$ and for each open set $U\subset U$ a certain space $S\_U$ which satisfies an appropriate sheaf condition ($U \mapsto S\_U$ is a sheaf of spaces). You want to prove that $S\_X$ is nonempty. Often, a small impr... | 2 | https://mathoverflow.net/users/9928 | 86086 | 51,205 |
https://mathoverflow.net/questions/86089 | 11 | When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
>
> Let $G$ be a topological group. A character of $G$ is a homomorphism of topological groups from $G$ to the torus $T=\mathbb{R}/\mathbb{Z}$. Here, the torus $T$ has the induced... | https://mathoverflow.net/users/39742 | Two Definitions of "Character" of topological groups | I am assuming all groups we are talking about are locally compact and commutative.
The two definitions indeed do ageree on profinite groups. To prove it, you have to check that the functors $Hom(-,\mathbb Q/ \mathbb Z)$ and $Hom(-,\mathbb R/ \mathbb Z)$ both transform limits of compact groups into colimits of discret... | 11 | https://mathoverflow.net/users/5952 | 86093 | 51,207 |
https://mathoverflow.net/questions/86103 | 1 | Is there any closed-form expression for the following integral:
$\int\_0^\infty \frac{1}{(1+a\_i s) \prod\_{j=1}^n (1+a\_j s)^k} ds $
where the ai are >0 and k is a positive integer. And, if k is not an integer?
Thank you
| https://mathoverflow.net/users/20732 | integral of a rational function (1+a_i s)^-1/prod((1+a_j s)^k) | This looks very much like a Lauricella Hypergeometric function: <http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series>
| 1 | https://mathoverflow.net/users/11142 | 86105 | 51,211 |
https://mathoverflow.net/questions/86107 | 6 | Let $X$ be a ringed space. Recall that an $\mathcal{O}\_X$-module $M$ is called *coherent* if it is of finite presentation and for every open $U \subseteq X$ and any integer $n \ge 1$, the kernel of every morphism of $\mathcal{O}\_U$-modules $\mathcal O\_U^{\oplus n} \to M|\_U$ is of finite type. Coherent modules const... | https://mathoverflow.net/users/2841 | Tensor product of coherent modules | In the analytic category this is indeed true: if $\mathscr{F}$, $\mathscr{G}$ are coherent analytic sheaves on a complex space $X$, then $\mathscr{F} \otimes\_{\mathscr{O}\_X} \mathscr{G}$ is also coherent.
For a reference, look at [Grauert-Remmert, Coherent Analytic Sheaves], Proposition at the bottom of page 240.
... | 8 | https://mathoverflow.net/users/7460 | 86108 | 51,213 |
https://mathoverflow.net/questions/86104 | 3 | A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms:
* $a\cdot(b\cdot x) = (ab)\cdot x$ for all reals $a$ and $b$, and all $x$ in $C$,
* $d(a\cdot x;a\cdot y) = \vert a\vert d(x;y)$ for all real $a$ and all... | https://mathoverflow.net/users/20733 | Independence of the axiomatics of metric cones | Yes, it is independent.
Consider function $f\colon \mathbb R^2\to \mathbb R\_\ge$ defened the following way:
$$f(x,y)=f(y,x)$$
and if $|y|\le |x|$ then
$$f(x,y)=|x|+\min\{|y|,|y-\tfrac12{\cdot} x|\}.$$
Note that one can define a metric on the union of two coordinate lines $C=\{(x,y)\in \mathbb R^2\mid x{\cdot}y=0\}$ ... | 5 | https://mathoverflow.net/users/1441 | 86109 | 51,214 |
https://mathoverflow.net/questions/85786 | 0 | im trying to determine a method to compare one particular time series against about 10,000+ reference time series programatically.. and shortlist those reference time series which can be of interest.. the method i was using was **Pearson Correlation**.. for each of the reference time series, i calculate their correlati... | https://mathoverflow.net/users/20603 | a reliable measure of series similarity - correlation just doesnt cut it for me.. | If you wish to stay model-free, then interquartile range and [rank correlation](http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html) are much more robust measures. Obviously with 10,000 candidate series [interquartile range](http://mathworld.wolfram.com/InterquartileRange.html) is too coarse a measure ... | 1 | https://mathoverflow.net/users/20665 | 86113 | 51,216 |
https://mathoverflow.net/questions/85511 | 0 | Can you use the Crank-Nicolson method to get a numerical approximation to the fisher-kolmogorov equation?
If not what would be the easiest way to model the equation using matlab?
Thanks and sorry its so basic but i'm new to these sorts of numerical approximations
| https://mathoverflow.net/users/20521 | Numerical solution to Fisher-Kolmogorov equation | Crank-Nicolson will work fine, though a fully implicit scheme may actually perform a bit better especially if you have discontinuous initial conditions. Since Crank-Nicolson is effectively half explicit, discontinuities in the initial conditions $u(x,0)$ propagate into a lot of noise echoing around your solution grid.
... | 0 | https://mathoverflow.net/users/20665 | 86115 | 51,218 |
https://mathoverflow.net/questions/86124 | 5 | I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read that the usual example is the blow-up of the plane in the $9$ points of intersection of $2$ general cubics. This should gi... | https://mathoverflow.net/users/1887 | Example of cone of numerically effective curves which is not polyhedral | Yes, this is the standard example of a variety whose cone of curves has infinitely many extremal rays. (A reference is Koll\'ar--Mori, p.22).
To see why there are infinitely many (-1)-curves: each of the 9 points you blow up gives a (-1)-curve E\_i, as you know. But the E\_i are also sections of the elliptic fibratio... | 6 | https://mathoverflow.net/users/nan | 86125 | 51,223 |
https://mathoverflow.net/questions/86120 | 15 | I keep reading that the Reshetikhin-Turaev construction actually yields a 3-2-1 tqft. I know the construction that associates to a suitably decorated surface a vector space built up from a hom-space in a modular tensor category and to a decorated 3-manifold a linear map between these vector spaces as described in the b... | https://mathoverflow.net/users/3995 | Reshetikhin-Turaev as a 3-2-1-theory | Hi Ulrich,
I think that's a good way to think about it. There's also a good reason why we have the associations
>
> *1-manifold <--> Linear Category*
>
>
> *2-manifolds F <--> Functors <--> Vector Spaces (when F is closed)*
>
>
> *3-manifolds X <--> Natural Transformations <--> Numbers (when X
> is closed),*... | 20 | https://mathoverflow.net/users/3593 | 86127 | 51,224 |
https://mathoverflow.net/questions/86003 | 11 | **Question:** Given a variety $X\_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber is normal?
My one thought is that perhaps this reduces to checking whether some 1-dimensional thing is smoothable... | https://mathoverflow.net/users/3521 | When is there a deformation of a given singularity to a normal singularity | Your $X\_0$ is Cohen-Macaulay of codimension $2$ in affine space, so determinantal (Hilbert-Burch). When also $\dim X\_0\le 3$ it is smoothable; see Schaps' paper in Am. J. Math., vol. $99$, for all this.
| 5 | https://mathoverflow.net/users/8726 | 86134 | 51,229 |
https://mathoverflow.net/questions/86099 | 4 | Suppose that the real algebraic curve $\gamma$ in $\mathbb{R}^3$ is the intersection of the zero sets of the polynomials $p\_1,...,p\_k \in \mathbb{R}[x,y,z]$. Is the projection of $\gamma$ on a generic plane (isomorphic to $\mathbb{R}^2$) a real algebraic curve? I.e., is it the zero set of one polynomial $p \in \mathb... | https://mathoverflow.net/users/19691 | Projections of real algebraic curves | As already mentioned, the answer is negative for varieties.
However, if you are interested in a more general setting when the answer is positive (i.e. a more general class of sets that *is* closed under projections), you can look at
* semialgebraic sets in the case of real closed fields (e.g. $\mathbb{R}$),
* const... | 3 | https://mathoverflow.net/users/20101 | 86155 | 51,240 |
https://mathoverflow.net/questions/86123 | 2 | Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D\_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection and simple normal crossings such that there is a morphism $f:S\rightarrow S'$ which collapses them to a point.
Can we f... | https://mathoverflow.net/users/1887 | Numerically negative exceptional divisor on a surface. | The intersection matrix $(D\_i\cdot D\_j)$ is negative definite and hence invertible. So the system of equations: $\sum\_i d\_i D\_i\cdot D\_j=\alpha\_j$ is solvable for any set of $\alpha\_j$'s, in particular for negative ones. It is an easy consequence of the negative definite property that if $\alpha\_j<0$ for all $... | 7 | https://mathoverflow.net/users/10076 | 86157 | 51,241 |
https://mathoverflow.net/questions/85828 | 2 | By "algebraic curve" here (both in $\mathbb{R}^3$ and in $\mathbb{C}^3$), I do not only mean that the dimension of the set as an algebraic variety is 1, but also that its dimension is 1 in a topological sense (i.e., that, roughly, it is locally a continuous parametric curve).
In this context, for example, the zero s... | https://mathoverflow.net/users/19691 | A real algebraic curve in $\mathbb{R}^3$ is the intersection of zero sets of polynomials in $\mathbb{R}[x,y,z]$. Can we choose the polynomials in a way, that, seen in $\mathbb{C}[x,y,z]$, the intersection of their zero sets is a complex alg. curve? | The "right" way to look at these is through algebra. In the case of the curve defined by $x^2+y^2=0$, what you want to extend to $\mathbb C$ is its *ideal*, not any particular equation. If you take $I=\{f\in\mathbb R[x,y,z]] | f(P)=0\text{ for every }P=(x,y,z), \text{ for which } x^2+y^2=0\}$, then you get exactly the ... | 3 | https://mathoverflow.net/users/10076 | 86158 | 51,242 |
https://mathoverflow.net/questions/86163 | 12 | I always believed the following statement: if $X$ is a smooth variety over an algebraically closed field of positive characteristic, assuming we know that the general member of a base point free linear system $|L|$ is reduced, then indeed a general member is smooth.
However, I realize this is not obvious, though all ... | https://mathoverflow.net/users/10083 | Bertini's theorem in char p for base point free linear system | In characteristic 3, consider the surface $V\subset \mathbb{P}^2\times\mathbb{A}^1$ with equation $y^2z=x^3-tz^3$ (where $t$ is the coordinate on $\mathbb{A}^1$). It is easily seen to be smooth. The general fiber of the projection on $\mathbb{A}^1$ is a plane cubic which is reduced but not smooth. Taking a suitable pro... | 17 | https://mathoverflow.net/users/7666 | 86181 | 51,252 |
https://mathoverflow.net/questions/84993 | 6 | Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(x)$ (and the same for $y$) don't fall in $k$, let call them $x^\sigma$ and $y^\sigma$. Then I can compute $k(x^\sigma, y... | https://mathoverflow.net/users/6776 | Computing the fixed field of an automorphism of a function field | [Colin Weir](http://math.ucalgary.ca/profiles/colin-weir), suggested the following algorithm to solve the problem in non-rational case, I thought for the sake of others who probably have the same question, I'll post it, here:
Suppose that $\sigma$ is an automorphism of $k(x,y)$. Using above theorem we can find a $x^\... | 0 | https://mathoverflow.net/users/6776 | 86189 | 51,256 |
https://mathoverflow.net/questions/86118 | 47 | Recall that a [derangement](http://en.wikipedia.org/wiki/Derangement) is a permutation $\pi: \{1,\ldots,n\} \to \{1,\ldots,n\}$ with no fixed points: $\pi(j) \neq j$ for all $j$. A classical application of the inclusion-exclusion principle tells us that out of all the $n!$ permutations, a proportion $1/e + o(1)$ of the... | https://mathoverflow.net/users/766 | Non-enumerative proof that there are many derangements? | 1. The mean number of fixed points is 1. This is very elementary.
2. Consider the operation of rotating three values around: $p(i)\to p(j)\to p(k)\to p(i)$. Given a permutation with no fixed points, there are $n^2-O(n)$ rotations that create from it a permutation with exactly one fixed point. Given a permutation with e... | 64 | https://mathoverflow.net/users/9025 | 86202 | 51,262 |
https://mathoverflow.net/questions/58861 | 5 | I know from classical category theory that if $C$ is a small category and $X$ is a presheaf, that there is a canonical equivalence $$Set^{C^{op}}/X \simeq Set^{\left(C/X\right)^{op}},$$ where $C/X$ is the category of elements of $X$ (i.e. the Grothendieck construction of $X$). Moreover, if $C$ carries a Grothendieck to... | https://mathoverflow.net/users/4528 | Slices of infinity sheaves | This is Corollary 5.1.6.12 in HTT. Somehow I overlooked this.
| 2 | https://mathoverflow.net/users/4528 | 86209 | 51,265 |
https://mathoverflow.net/questions/86208 | 3 | How can we deduce the three dimensional spherical space form conjecture from the Poincare conjecture? More precisely, how can we deduce using the Poincare conjecture that every free action of a finite group on $\mathbb{S}^3$ is equivalent to an orthogonal group action. If the proof is involved, then kindly suggest some... | https://mathoverflow.net/users/20620 | Three dimensional spherical space form | You can't deduce this from the Poincare conjecture, but it follows from the geometrization conjecture/theorems for manifolds and orbifolds. I assume (though I haven't checked) [Morgan-Tian](https://arxiv.org/abs/0809.4040) talk about this.
| 3 | https://mathoverflow.net/users/11142 | 86214 | 51,269 |
https://mathoverflow.net/questions/86022 | 16 | Suppose $(X^{2n},\omega)$ is a compact symplectic manifold. Knowing the algebra $C^\infty(X)$ is equivalent to knowing the manifold $X$, and knowing the Poisson bracket $\{\cdot,\cdot\}:C^\infty(X)\otimes C^\infty(X)\to C^\infty(X)$ is equivalent to knowing $\omega$. Thus it is natural to ask the following question:
... | https://mathoverflow.net/users/35353 | Can you tell the volume of a symplectic manifold from the Poisson brackets? | The Poisson geometry is, I think, a red herring. I will explain in this answer how to construct $\Omega^{2n}(X)$ as a $C^{\infty}(X)$ module, and how to construct the volume form $\omega^{n}$ within it. So I think the right question to ask is:
>
> Let $X^m$ be a smooth, orientable, compact manifold. Suppose that we... | 7 | https://mathoverflow.net/users/297 | 86215 | 51,270 |
https://mathoverflow.net/questions/86197 | 14 | We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question :
Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g \neq 0, \Delta (g) = g \otimes g$ } the set of group-like elements. We know that this set is a monoid and that if B has... | https://mathoverflow.net/users/59028 | Is a bialgebra with all group-like elements invertible a Hopf algebra? | The answer is "no". A counterexample is given by Radford in Example 2 (p. 567) in the paper
$\quad\quad$[On Bialgebras which are simple Hopf Modules, Amer. Math. Soc. 80(1980),563-568](https://www.ams.org/journals/proc/1980-080-04/S0002-9939-1980-0587928-4/S0002-9939-1980-0587928-4.pdf)
He takes the coalgera $C = \... | 15 | https://mathoverflow.net/users/10194 | 86231 | 51,273 |
https://mathoverflow.net/questions/86191 | 8 | I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
1. Here X can be even any Tychonoff space. Is there a description of the dual space of C0(X) as measures, or some close theorem? I have fo... | https://mathoverflow.net/users/19471 | Measures on general topological groups | There are some results on the representation of certain functionals by measures in the paper
Smolyanov, O.G.; Fomin, S.V. Measures on linear topological spaces. Russ. Math. Surv. 31, No.4, 1-53 (1976); translation from Usp. Mat. Nauk 31, No.4(190), 3-56 (1976).
In fact, without local compactness the problem acquir... | 4 | https://mathoverflow.net/users/12205 | 86242 | 51,278 |
https://mathoverflow.net/questions/86173 | 5 | (Crossposted from [math.SE](https://math.stackexchange.com/questions/99784/morita-equivalence-of-acyclic-categories).)
Call a category *acyclic* if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. Suppose that they are *Mor... | https://mathoverflow.net/users/290 | Morita equivalence of acyclic categories | The answer to your question is "no". A counterexample is given in this [paper](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-26/issue-1/The-isomorphism-problem-for-incidence-algebras-of-M%C3%B6bius-categories/10.1215/ijm/1256046901.full) by Leroux in Example 1.6.
| 2 | https://mathoverflow.net/users/15934 | 86244 | 51,279 |
https://mathoverflow.net/questions/86243 | 6 | For a convex polytope $P$ in $\mathbb R^4$, denote by $N\_0,N\_1,N\_2,N\_3$ respectively the number of vertices, edges, faces, cells. By Euler's formula, we know $N\_0+N\_2=N\_1+N\_3$, which means there is a sort of equilibrum among the $N\_i$. But I wonder if there exist upper and/or lower bounds for $f(P):=\frac{N\_1... | https://mathoverflow.net/users/29783 | convex polytopes with many faces and edges but few cells and vertices | As of 2003, I don't think it was known whether $f(P)$ is bounded. See *Fat 4-polytopes and fatter 3-spheres* by Eppstein, Kuperberg, and Ziegler (Monogr. Textbooks Pure Appl. Math. **253** (2003), 239-265, doi:[10.1201/9780203911211](https://doi.org/10.1201/9780203911211), arxiv:[math/0204007](https://arxiv.org/abs/mat... | 7 | https://mathoverflow.net/users/4720 | 86245 | 51,280 |
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