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https://mathoverflow.net/questions/20423 | 4 | Given a Ferrers board of shape $(b\_1,\ldots,b\_m)$, we define $r\_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's shown the identity:
$$\sum\_k r\_k (x)\_{m-k} = \prod\_i (x+s\_i)$$
where $s\_i = b\_i-i+1$, but I don't know if I c... | https://mathoverflow.net/users/961 | How to compute the rook polynomial of a Ferrers board? | You can retrieve the coefficients of a polynomial written in the falling factorial basis by computing finite differences, as follows.
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function, and let $\Delta f(n) = f(n+1) - f(n)$. Let $\Delta^{r+1} f = \Delta(\Delta^r f)$.
**Lemma 1:** $\displaystyle \Delta {n \choose k}... | 6 | https://mathoverflow.net/users/290 | 20464 | 13,593 |
https://mathoverflow.net/questions/20444 | 11 | Let $K$ be a field and $n \geq 1$. Then the set of isomorphism classes of vector bundles over $\mathbb{P}^n\_K$ is a semiring (i.e. almost a ring, but no additive inverses are possible). By introducing additive inverses and quotienting out exact sequences, we get the $K$-theory of $\mathbb{P}^n\_K$, which is known to b... | https://mathoverflow.net/users/2841 | Semiring of algebraic vector bundles on projective space | This semiring carries an enourmous amount of information about vector bundles on $\mathbb{P}^n$, including stuff we don't yet know. For example, you can read from it whether there are indecomposable vector bundles of any given rank; and for small rank we know very little about it (this is discussed, for example, in C. ... | 10 | https://mathoverflow.net/users/4790 | 20470 | 13,597 |
https://mathoverflow.net/questions/20442 | 13 | A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with respect to maps $Y\times \{0\} \to Y \times [0,1]$ such that the lift agrees with the map $Y \to E$ up to a vertical homotop... | https://mathoverflow.net/users/4177 | Request: A Serre fibration that is not a Dold fibration | (answering my own question - who would have thought?)
There is a paper by G Allaud (Arch. Math 1968) which describes a counterexample as sought by the question. Let $E$ be the subspace of the plane $\mathbb{R}^2$ consisting of the non-negative integer points $(n,0)$ on the $x$-axis together with $(0,1)$ and a line c... | 9 | https://mathoverflow.net/users/4177 | 20479 | 13,602 |
https://mathoverflow.net/questions/20478 | 3 | What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example [this](http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/29/paper_html/node10.html). For matrix factorizations, see for example... | https://mathoverflow.net/users/83 | Matrix factorization categories for ADE singularities | See: "Matrix Factorizations and Representations of Quivers II: type ADE case" (math/0511155) by Kajiura, Saito, and Takahashi for a recent account.
Older references include:
"Construction geometrique de la correspondance de McKay" Gonzalez-Sprinberg,and Verdier (1983)
Y. Yoshino, Cohen-Macaulay modules over Cohen-Mac... | 2 | https://mathoverflow.net/users/874 | 20481 | 13,603 |
https://mathoverflow.net/questions/20471 | 41 | Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a group to its profinite completion injective if the group is free?
Apparently, this follows from a result of Malcev's t... | https://mathoverflow.net/users/1474 | Why are free groups residually finite? | Here is a direct proof for free groups.
Let $x\_1,\dots,x\_m$ be the generators of our group. Consider a word $x\_{i\_n}^{e\_n}\dots x\_{i\_2}^{e\_2}x\_{i\_1}^{e\_1}$ where $e\_i\in\{\pm 1\}$ and there are no cancellations (that is, $e\_k=e\_{k+1}$ if $i\_k=i\_{k+1}$).
I'm going to map this word to a nontrivial ele... | 42 | https://mathoverflow.net/users/4354 | 20485 | 13,606 |
https://mathoverflow.net/questions/20493 | 209 | Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown\_X Y - \triangledown\_Y X - [X,Y]$$
What is the geometric picture behind this definition—what does torsion measure intuitively?
| https://mathoverflow.net/users/2837 | What is torsion in differential geometry intuitively? | The torsion is a notoriously slippery concept. Personally I think the best way to understand it is to generalize past the place people first learn about torsion, which is usually in the context of Riemannian manifolds. Then you can see that the torsion can be understood as a sort of obstruction to integrability. Let me... | 137 | https://mathoverflow.net/users/184 | 20506 | 13,621 |
https://mathoverflow.net/questions/20503 | 3 | Here is an exercise from Serre's "local fiels" when he starts to do cohomology: Let G act on an abelian group A, f be an inhomogenous n cochain, i.e. $f\in C^n(G,A).$ Define an operator T on f, $Tf(g\_1,g\_2,\cdots,g\_n)=g\_1g\_2\ldots g\_n f(g\_n^{-1},g\_{n-1}^{-1},\ldots,g\_1^{-1})$. It is clear that $T^2f=f$. It is ... | https://mathoverflow.net/users/1877 | An exercise in group cohomology | Fix the signs as Wilberd suggests in the comments, check that you get a natural automorphism of the complex computing cohomology, see that it induces in fact an automorphism of the universal $\delta$-functor $H^\bullet(G,\mathord-)$, and see what it does in degree zero.
| 4 | https://mathoverflow.net/users/1409 | 20509 | 13,623 |
https://mathoverflow.net/questions/20512 | 2 | Let A be a borelian set with postivie measure. I was asking myself if it is possible to find an open set $B\subseteq A$ such that $B$ is an open set minus a set of null measure...
| https://mathoverflow.net/users/4928 | If a Borelian set has positive measure, does it contain a non empty open set (minus a measure null set)? | The Cantor set of positive measure is nowhere dense set. So it is an example.
| 11 | https://mathoverflow.net/users/2823 | 20513 | 13,625 |
https://mathoverflow.net/questions/20516 | 12 | I'm having trouble distinguishing the various sorts of tori.
One definition of torus is the algebraic torus. Groups like $SU(2,\mathbb{C})$ and $SU(3,\mathbb{C})$ have important subgroups that are topologically a circle and a torus, and I guess those were some of the most important Lie groups so the name torus stuck.... | https://mathoverflow.net/users/3710 | Complex torus, C^n/Λ versus (C*)^n | The difference between an Abelian variety and $\mathbb{C}^n/\Lambda$ is that an abelian variety is *polarized*; that is, it comes with an ample line bundle, which yields an embedding into $\mathbb{P}^m$ for some $m$.
That is, Abelian varieties are projective algebraic, whereas complex tori (in the sense of $\mathbb{C... | 13 | https://mathoverflow.net/users/1703 | 20519 | 13,627 |
https://mathoverflow.net/questions/20511 | 16 | In chapter 13 in [his notes on 3-manifolds](http://www.msri.org/communications/books/gt3m/PDF/13.pdf), Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "Later we shall interpret $\pi\_1(O)$ in terms of loops on $O$,... | https://mathoverflow.net/users/353 | Orbifold fundamental group in terms of loops? | Most of the standard intro sources on orbifolds discuss their fundamental groups in terms of coverings. One exception is Ratcliffe's book "Foundations of Hyperbolic Manifolds", chapter 13 of which contains a discussion of the fundamental group of an orbifold defined via loops.
| 12 | https://mathoverflow.net/users/317 | 20521 | 13,629 |
https://mathoverflow.net/questions/20507 | 10 | Let $S$ be a subset of $\mathbb{R}^n$ defined by a system $\theta$ of polynomial inequalities with integer coefficients. Let $S+\mathbb{Z}^n$ be all points of the form $s+z$ with $s \in S$ and $z \in \mathbb{Z}^n$. Is there a method known for determining, given $\theta$, whether $S+\mathbb{Z}^n=\mathbb{R}^n$? Is this p... | https://mathoverflow.net/users/5229 | A Decision Problem Concerning Diophantine Inequalities | It is undecidable. If you could solve this, you could also solve [Hilbert's 10th problem](http://en.wikipedia.org/wiki/Hilbert%2527s_tenth_problem).
Suppose we have an algorithm solving your problem for all $n$. Given a polynomial $p\in\mathbb[x\_1,\dots,x\_n]$, let's decide whether it has integer solutions. If $p$ i... | 9 | https://mathoverflow.net/users/4354 | 20522 | 13,630 |
https://mathoverflow.net/questions/20453 | 20 | What is the relationship between $G\_\infty$ (homotopy Gerstenhaber) and $B\_\infty$ algebras?
In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper I don't well understand) a $B\_\infty$ algebra is defined to be a graded vector space $V$ together with a dg-bialgebra s... | https://mathoverflow.net/users/132 | Are G_infinity algebras B_infinity? Vice versa? | There is a nice summary of the relationship between B infinity and G infinity in the first chapter of the book "Operads in Algebra, Topology and Physics" by Markl, Stasheff and Schnider. The short answer is G infinity is the minimal model for the homology of the little disks operad (the G operad). B infinity is an oper... | 9 | https://mathoverflow.net/users/4960 | 20530 | 13,635 |
https://mathoverflow.net/questions/20531 | 4 | Consider the usual language and axioms of ZF. Now add constants $x\_1, x\_2, \dots$ to the language together with the axioms $x\_2\in x\_1, x\_3\in x\_2, \dots$ to form a new theory. Then by the compactness theorem, since every finite subset of the axioms has a model, the new theory has a model. But doesn't the set {$x... | https://mathoverflow.net/users/416 | Contradiction to axiom of foundation | You have already answered your question yourself: the set $\{x\_1, x\_2, \ldots\}$ cannot exist in the model, since this would violate the Foundation Axiom. You cannot get this set from Replacement, since there is no definable function having this set as its range, required to invoke the Replacement Axiom.
What you ... | 11 | https://mathoverflow.net/users/1946 | 20532 | 13,636 |
https://mathoverflow.net/questions/20538 | 9 | Let $L$ be a lattice in $\mathbb{C}$ with two fundamental periods, so that $\mathbb{C}/L$ is topologically a torus. Let $p:\mathbb{C}/L \mapsto \mathbb{R}^3$ be an embedding ($C^1$, say). Call $p$ conformal if pulling back the standard metric on $\mathbb{R}^3$ along $p$ yields a metric in the equivalence class of metri... | https://mathoverflow.net/users/5181 | conformally embedding complex tori into R^3 | You should have a look in Pinkall's Hopf Tori paper. You take the preimage of a curve in $S^2$ under the Hopf fibration. The lattice of the torus and hence the conformal class is then given by the generators $1\in C$ and $L+i/2 A$ (if I remember right), where $L$ is the length and $A$ is the enclosed area of the curve.... | 11 | https://mathoverflow.net/users/4572 | 20540 | 13,640 |
https://mathoverflow.net/questions/20539 | 10 | The conjugacy problem for a free group $F\_n$ on $n$ letters has an easy solution. Each element of $F\_n$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if you arrange the word around a circle, then there are no cancellations), so two elements of $F\_n$ are conjugate if and... | https://mathoverflow.net/users/4682 | Decidability of conjugacy problem for finitely generated subgroups of free groups | There is an algorithm to do this. I would have thought that it was classical, but in any case an algorithm is given in: I. Kapovich and A. Myasnikov "Stallings foldings and the subgroup structure of free groups", J. Algebra 248 (2002), no 2, pp. 608-668. In the online version I found [here](http://www.math.uiuc.edu/~ka... | 7 | https://mathoverflow.net/users/1109 | 20544 | 13,643 |
https://mathoverflow.net/questions/20551 | 49 | Does anyone know of a good place to find already-done BibTeX entries for standard books in advanced math? Or is this impossible because the citation should include items specific to your copy? (I am seeing the latter as potentially problematic because the only date I can find in my copy of Hartshorne is 2006, whereas t... | https://mathoverflow.net/users/5094 | Sources for BibTeX entries | I recommend using the AMS website MREF, located [here](https://mathscinet.ams.org/mref).
EDIT : Another remark about your question. Don't worry too much about getting things like the printing date for a book correct (it changes every time they make a new printing run). Just make sure that you have the author, title, ... | 41 | https://mathoverflow.net/users/317 | 20552 | 13,648 |
https://mathoverflow.net/questions/20548 | 6 | For example, how to solve the equation $\sum^{p-1}\_{i}x\_{i}^{2}=0$ in $F\_{p}$? This is not a homework problem. I think it should have a definite answer, so not an open problem. I just don't know how to solve it.
| https://mathoverflow.net/users/5175 | How to solve Diophantine equations in $F_{p}$? | There is a *deterministic* polynomial-time algorithm for finding solutions to diagonal equations of degree less than or equal to the number of variables over finite fields. See [Christiaan van de Woestijne's thesis](http://www.opt.math.tugraz.at/~cvdwoest/).
(A solution of your example equation can be found much more... | 14 | https://mathoverflow.net/users/2757 | 20553 | 13,649 |
https://mathoverflow.net/questions/20534 | 8 | Related to the question [link text](https://mathoverflow.net/questions/18636/number-of-invertible-0-1-real-matrices) I was asking myself some time ago the following. Can one precisely describe the invertible n\times n matrices with{0, 1} entries? For example, is anything special about the graph asociated to this matrix... | https://mathoverflow.net/users/5179 | Characterizing invertible matrices with {0,1} entries | You are unlikely to find a characterization which does not result from simple facts in linear algebra. I am unaware of any characterizations which make interesting statements about graphs.
You may want to choose the ring over which the matrices belong. For example, the same matrix may be invertible over the reals, bu... | 6 | https://mathoverflow.net/users/3402 | 20559 | 13,654 |
https://mathoverflow.net/questions/20529 | 3 | Edit: Rewritten with motivation, and hopefully more clarity.
I'm building a site for a card game called [dominion](http://www.boardgamegeek.com/boardgame/36218/dominion). In it, people build 'decks' of 10 distinct cards from a set of (currently) approximately 80. People (will) upload their decks to the site I'm worki... | https://mathoverflow.net/users/4379 | Random generation of subsets using conditional probabilities | You may do better with an approach that mimics the likely characteristics, and then selects cards that meet the characteristics, and then resolves conflicts. Here is a possible approach:
Consider the gross characteristics of such a deck: number and distribution of costs, number of +n Buys +n Cards +n Actions, number ... | 0 | https://mathoverflow.net/users/3402 | 20569 | 13,661 |
https://mathoverflow.net/questions/20568 | 12 | It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started in topology: namely, whether $\mathbb{R}$ is the only complete dense unbounded linearly ordered set that satisfies the c... | https://mathoverflow.net/users/3859 | Is there a ground between Set Theory and Group Theory/Algebra? | Descriptive set theory also has something to say about algebra ... For example, the Higman-Neumann-Neumann Embedding Theorem
states that any countable group G can be embedded into a
2-generator group K. In the standard proof of this classical
theorem, the construction of the group K involves an
enumeration of a set of ... | 14 | https://mathoverflow.net/users/4706 | 20574 | 13,665 |
https://mathoverflow.net/questions/19907 | 5 | Is there much known about the theory of lax and colax monads on a bicategory? Here, I really mean lax or colax, not weak. I'm aware of some literature about weak monads. I'm interested in distributive laws of lax and colax monads and their relationships with algebras. I've managed to prove some things, but, I don't wan... | https://mathoverflow.net/users/4528 | Lax and Colax Monads | This isn't technically an answer, but depending on your examples, you might want to think about lax/colax monads on (pseudo) double categories instead. Part of the problem with lax monads on bicategories is that there is no tricategory of bicategories and lax functors, whereas there is a 2-category of pseudo double cat... | 2 | https://mathoverflow.net/users/49 | 20575 | 13,666 |
https://mathoverflow.net/questions/19011 | 2 | Assume I have a set of weighted samples, where each samples has a corresponding weight between 0 and 1. I'd like to estimate the parameters of a gaussian mixture distribution that is biased towards the samples with higher weight. In the usual non-weighted case gaussian mixture estimation is done via the EM algorithm. D... | https://mathoverflow.net/users/4806 | Estimate gaussian (mixture) density from a set of weighted samples | The usual EM algorithm can be modified for weighted inputs. Following along the [Wikipedia presentation](http://en.wikipedia.org/wiki/Mixture_model#Expectation_maximization_.28EM.29), you would use these formulas instead:
$a\_i = \frac{\sum\_{j=1}^N w\_j y\_{i,j}}{\sum\_{j=1}^{N}w\_j}$
and
$\mu\_{i} = \frac{\sum\... | 2 | https://mathoverflow.net/users/634 | 20579 | 13,670 |
https://mathoverflow.net/questions/20580 | 9 | This is a follow-up question to this [coend computation](https://mathoverflow.net/questions/20445/coend-computation). Here's an attempt at a slightly simpler computation:
>
> $\int^{a \in A} \mbox{hom}\_A(a,a)$
>
>
>
This should be similar to the trace operator. In attempting to follow the derivation
>
> $... | https://mathoverflow.net/users/756 | Coend computation continued | I agree with Reid's answer, but I want to add a bit more.
Putting Reid's calculation into a more general setting, if $A$ is *any* category then
$$
\int^{a \in A} \mathrm{hom}\_A (a, a)
= (\mathrm{endomorphisms\ in\ } A)/\sim
$$
where $\sim$ is the (rather nontrivial) equivalence relation generated by $gh \sim hg$ wh... | 8 | https://mathoverflow.net/users/586 | 20592 | 13,678 |
https://mathoverflow.net/questions/20549 | 8 | Given a character $\chi$ and $k$ odd how can one compute a basis for the space of modular forms $M\_\frac{k}{2}(\Gamma\_0(4),\chi)$. By compute a basis I mean, finding the beginning of the Fourier expansions. I am looking for computer programs, which can do that for me.
I have heard of the package SAGE, which seems t... | https://mathoverflow.net/users/3757 | Basis for modular forms of half-integral weight | **Edit:** Here's a rather silly method that should work if SAGE is just giving you cusp forms: $\Gamma\_0(4)$ has a single normalized cusp form of weight 6, given by $\eta(2\tau)^{12} = q - 12q^3 + 54q^5 - \dots$, so take your basis of cusp forms of weight $k/2 + 6$, and divide each element by this form to get a basis ... | 7 | https://mathoverflow.net/users/121 | 20595 | 13,679 |
https://mathoverflow.net/questions/20590 | 11 | Here's the setup: you have a first-order theory T, in a countable language L for simplicity. Let k be a cardinal and suppose T is k-categorical. This means that, for any two models
M,N |= T
of cardinality k, there is an isomorphism f : M --> N.
Supposing all this happens inside of ZFC, let's say I change the unde... | https://mathoverflow.net/users/4367 | How does categoricity interact with the underlying set theory? | Categoricity is absolute.
By the Ryll-Nardzewski theorem, for a countable language, $\aleph\_0$-categoricity of a complete theory $T$ is equivalent to $T$ proving for each natural number $n$ that there are only finitely many inequivalent formulas in $n$ variables. This property is evidently arithmetic and, thus, abs... | 22 | https://mathoverflow.net/users/5147 | 20601 | 13,681 |
https://mathoverflow.net/questions/20609 | 19 | When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point:
$$\det \left(\frac{1}{x \_i+y \_j}\right) \_{1\le i,j \le n}=\frac{\prod \_{1\le i < j\le n} (x \_j-x \_i)(y \_j-y \_i)}{\prod \_{i,j=1}^n (x \_i+y \_j)}$$
This goes under... | https://mathoverflow.net/users/2384 | What role does Cauchy's determinant identity play in combinatorics? | See also pp. 397--398 of *Enumerative Combinatorics*, vol. 2. Cauchy's determinant is given in a slightly different but equivalent form. It is explained there that the evaluation of the determinant is equivalent to the fundamental identity $\prod(1-x\_iy\_j)^{-1} =\sum\_\lambda s\_\lambda(x)s\_\lambda(y)$ in the theory... | 16 | https://mathoverflow.net/users/2807 | 20639 | 13,706 |
https://mathoverflow.net/questions/20645 | 3 | Let $E$ and $F$ be two abelian varieties of dimension 1 over $\mathbb{C}$. Let $f : E \to F$ be a surjective homomorphism of abelian varieties ($f(0) = 0$). If $\ker (f) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, does this imply that $E$ and $F$ are isomorphic?
| https://mathoverflow.net/users/5197 | isogeny of elliptic curves | Yes. In zero characteristic the image of an isogeny of elliptic curves
is determined up to isomorphism by its kernel. Your isogeny has the same kernel
as the doubling map from $E$ to itself.
| 12 | https://mathoverflow.net/users/4213 | 20647 | 13,710 |
https://mathoverflow.net/questions/20608 | 1 | I have the following setup:
$X, Y$ are topological spaces (if it helps, they can both be $T\_1$ and normal. They can even be countably paracompact. They *can't* be assumed paracompact). $V$ is a normed space (it can be Banach if you like). $f : X \to Y$ is a perfect surjection.
I have continuous and bounded $g : X ... | https://mathoverflow.net/users/4959 | Approximate selection theorems for factoring through perfect maps | Consider $X=Y=S^1$. Let $f:X\to Y$ be a 2-fold covering and $g:X\to\mathbb R^2$ the standard embedding (whose image is a unit circle). Assume $\epsilon<1$, then there is no map $h$ with the desired property.
Indeed, if $d(h(x),g(f^{-1}(x)))<\epsilon$, then there is a unique $y\in f^{-1}(x)$ such that $d(h(x),g(y))<\e... | 4 | https://mathoverflow.net/users/4354 | 20653 | 13,714 |
https://mathoverflow.net/questions/20604 | 21 | The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems that mathematicians have always striven to "put the negatives" into whatever algebraic structure they came across, in anal... | https://mathoverflow.net/users/4721 | Are rings really more fundamental objects than semi-rings? | Of course the real question is whether abelian groups are really more fundamental objects than commutative monoids. In a sense, the answer is obviously no: the definition of commutative monoid is simpler and admits alternative descriptions such as the one I give [here](https://mathoverflow.net/questions/2551/why-do-gro... | 10 | https://mathoverflow.net/users/126667 | 20654 | 13,715 |
https://mathoverflow.net/questions/20651 | 9 | Hello everyone;
i'm looking for a motivation for equivariant sheaves (see <http://ncatlab.org/nlab/show/equivariant+sheaf>) ~ **Why are we interested in them?**
More explicitely: Can I think of G-equivariant sheaves on a space X as a quotient of the category of sheaves (by some action? in a more general sense) by G... | https://mathoverflow.net/users/1261 | Motivation for equivariant sheaves? | If you know that the sections of a vector bundle form a standard example of a sheaf, then the corresponding example of a $G$-equivariant sheaf on a space $X$ with $G$-action is a vector bundle $V$ over $X$ with a $G$-action compatible with the projection (i.e. making the projection $G$-equivariant, i.e. intertwining th... | 16 | https://mathoverflow.net/users/35833 | 20662 | 13,721 |
https://mathoverflow.net/questions/20634 | 8 | From the [Peter–Weyl theorem in Wikipedia](https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem), this theorem applies for compact group. I wonder whether there is a non-compact version for this theorem.
I suspect it because the proof of the Peter–Weyl theorem heavily depends on the compactness of Lie group. It i... | https://mathoverflow.net/users/1851 | Is there analogue of Peter–Weyl theorem for non-compact or quantum group |
>
> What I want to ask is there any other
> way to define quantized flag variety?
> In the classical case, it is well
> known that flag variety can be defined
> as $G/B$, say $G$ is general linear group
> and B is Borel subgroup. Is there any
> analogue for quantum case? Is there a
> definition like $G\_q$ as "quantu... | 5 | https://mathoverflow.net/users/35833 | 20668 | 13,724 |
https://mathoverflow.net/questions/20613 | 12 | Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that $A \subseteq \overline{B}(c, r)$?
The answer is clearly yes for finite dimensional normed spaces: Define $T\_\epsilon = \b... | https://mathoverflow.net/users/4959 | Radii and centers in Banach spaces | I believe that the property does not hold for all Banach spaces, but my counterexample is a little involved. If you've the patience then follow me through...
Let $V=\bigoplus\_{n=1}^\infty \ell^n\_2$ where $\ell^p\_2$ is $\mathbb{R}^2$ with
norm $\lVert\cdot\rVert\_p$ (Note: $n$ is taking the role of $p$). For $i\geq... | 7 | https://mathoverflow.net/users/5213 | 20672 | 13,727 |
https://mathoverflow.net/questions/20650 | 12 | Let $X$ be a Riemannian manifold. If $X$ is simply connected, irreducible, and not a symmetric space then we know that the possible holonomy groups of the metric on $X$ are:
1) $O(n)$ General Riemannian manifolds
2) $SO(n)$ Orientable manifolds
3) $U(n)$ Kahler manifolds
4) $Sp(n)Sp(1)$ Quaternionic Kahler ma... | https://mathoverflow.net/users/5124 | Special Holonomy Groups for Lorentzian Manifolds | I think it would be more accurate to say that the real reason why
Calabi-Yau, hyperkähler, $G\_2$ and $\mathrm{Spin}(7)$ manifolds are of
interest in string theory is not their Ricci-flatness, but the fact
that they admit parallel spinor fields. Of course, in
positive-definite signature, existence of parallel spinor fi... | 16 | https://mathoverflow.net/users/394 | 20673 | 13,728 |
https://mathoverflow.net/questions/20666 | 8 | The following question came up in the course on Quantum Groups here at UC Berkeley. (If you care, I have been TeXing [uneditted lecture notes](http://math.berkeley.edu/~theojf/QuantumGroups10.pdf).)
Let $X,Y$ be (infinite-dimensional) [Hopf algebras](http://en.wikipedia.org/wiki/Hopf_algebra) over a ground field $\ma... | https://mathoverflow.net/users/78 | Is a bialgebra pairing of Hopf algebras automatically a Hopf pairing? | Theo,
I think one can argue like this in the case of a non-degenerate pairing. I didn't check everything here carefully, so don't believe it unless you confirm it yourself.
One has from the pairing an inclusion of $i:X\hookrightarrow Y^\*$. One has two maps on $i(X)$, $S\_X$, and $S\_Y^\*|\_{i(X)}$. Both of these s... | 3 | https://mathoverflow.net/users/1040 | 20678 | 13,731 |
https://mathoverflow.net/questions/20675 | 16 | Katzarkov-Kontsevich-Pantev define a *smooth* dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is *perfect* if the functor $Hom(M,-)$ from ($A$-modules) to ($\mathbb{C}$-modules) preserves small homotopy colimits. (Definition 2.23 of [KKP](ht... | https://mathoverflow.net/users/83 | Smooth dg algebras (and perfect dg modules and compact dg modules) | The condition of Hom(M,-) being a continuous functor, i.e. preserving (small homotopy) colimits is equivalent (in the present stable setting) to the maybe more concrete condition of preserving arbitrary direct sums. The issue is not finite direct sums, that's automatic (since the derived Hom is an exact functor, it aut... | 27 | https://mathoverflow.net/users/582 | 20679 | 13,732 |
https://mathoverflow.net/questions/20615 | 5 | I have a lemma about antichains that I think should be already known, but I can't find it anywhere. I am looking for a reference to this result that I can use in my paper, so that I don't have to include the proof.
Let $\mathcal{F}$ be an antichain on finite universe $U$, such that there are no $m$ distinct subsets $... | https://mathoverflow.net/users/5200 | Maximum size of antichain if no m subsets have a common intersection of size n | Note that the interesting case is if $n \leq U/2$. Otherwise, your bound is worse than the bound $\binom{U}{\lfloor U/2 \rfloor}$ given by [Sperner's Theorem](http://en.wikipedia.org/wiki/Sperner_family), for the size of *any* antichain on the universe $U$. So assuming that $n \leq U/2$, we can improve the bound in (1)... | 1 | https://mathoverflow.net/users/2233 | 20680 | 13,733 |
https://mathoverflow.net/questions/20667 | 8 | The Original Fitch Cheney puzzle goes like this:
>
> A Volunteer from the crowd chooses any
> five cards at random from a deck, and
> hands them to you so that nobody else
> can see them. You glance at them
> briefly, and hand one card bakc, which
> the volunteer then places face down on
> the table to one si... | https://mathoverflow.net/users/2655 | Generalization of Finch Cheney's 5 Card Trick | Kleber's paper will certainly point you in the right direction if you can find it. (I have a printed out copy, and I don't remember where on the web I got it. Sorry.)
In it, he suggests thinking about a strategy as a pairing up of the $(n!+n-1)\\_{n-1}$ messages with the ${{n!+n-1}\choose n} = (n!+n-1)\\_{n-1}$ hands... | 6 | https://mathoverflow.net/users/1060 | 20735 | 13,763 |
https://mathoverflow.net/questions/20692 | 4 | Let $G= (\mathbb{Z} \bigoplus \mathbb{Z}) \star (\mathbb{Z} \bigoplus \mathbb{Z})$, where $\star$ denotes the free product, let F be the commutator subgroup of G, it is free by a theorem of Kurosh. Find a proper normal subgroup of F (other than the trivial one) such that it is of infinite index.
| https://mathoverflow.net/users/5218 | Finite index normal subgroups of a free group. | The commutator subgroup $F' = [F:F]$ of $F$. It is normal. $F$ is not abelian, so $F'$ is nontrivial. The quotient $F/F'$ is a free abelian group of infinite rank, so $[F:F']$ is infinite.
| 2 | https://mathoverflow.net/users/1149 | 20743 | 13,768 |
https://mathoverflow.net/questions/20750 | 4 | I remember to have seen a big list in the EGA of properties $(P)$ such that:
if $f : X \to Y$ has $(P)$ then, $f\_{(S')} : X\_{(S')}\to Y\_{(S')}$ has $(P)$, where $f\_{(S')}$ is the morphism $f$ after a base change $S'\to S$., etc. but I can't find it now...
Does anyone know where I can find such a list ?
(I am in... | https://mathoverflow.net/users/2330 | Properties stable under base change in algebraic geometry | One list I've seen is in Appendix C from a course 'Rational Points on Varieties' taught by Bjorn Poonen. Here's the link:
<http://math.mit.edu/~poonen/papers/Qpoints.pdf>
Also, the appendix to the book of Gortz and Wedhorn 'Algebraic Geometry 1: Schemes With Examples and Exercises' is a great reference.
| 15 | https://mathoverflow.net/users/386 | 20751 | 13,771 |
https://mathoverflow.net/questions/20646 | 4 | Are there any known reversible pairing functions $f: \mathbb N \times \mathbb N \to \mathbb N$ that can be computed in constant time (FAC⁰)?
| https://mathoverflow.net/users/2644 | Are there any pairing functions computable in constant time (AC⁰) | Interleaving the binary encodings of the two numbers a and b seems to be the best solution:
For example the encoding of
a = 20d = 10100b
b = 5d = 101b
We interleave the bits starting with the least significant bits (we pad shorter numbers with 0's so they are the same length).
The resulting paired number... | 5 | https://mathoverflow.net/users/2644 | 20755 | 13,775 |
https://mathoverflow.net/questions/20702 | 9 | I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
* vector bundles
* torsion-free sheaves
* principal bundles
* parabolic bundles
over *singular* algebraic curves (reducible or not), in any of the following frameworks:
* algebraic geometry (i... | https://mathoverflow.net/users/4721 | Reference request: Moduli spaces of bundles over singular curves | Some of the many (semi)standard references are below (with no claims to completeness or representativeness, if that's a word -- just the first references that came to mind). My feeling is the subject is still very much in its infancy however, for example one would like to know the standard package of nonabelian Hodge t... | 9 | https://mathoverflow.net/users/582 | 20758 | 13,778 |
https://mathoverflow.net/questions/20712 | 24 | Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not **homotopic** to a non-surjective map?
**Added**: The motivation is: There is a "mild version" of the Nearby Langrangian conjecture stating: any exact Lagrangian manif... | https://mathoverflow.net/users/4500 | Do "surjective" degree zero maps exist? | It is a theorem of H. Hopf that a map between connected, closed, orientable n-manifolds of degree 0 is homotopic to a map that misses a point, when n > 2. See D. B. A. Epstein, The degree of a map. Proc. London Math. Soc. (3) 16 1966 369--383, for a "modern" discussion including the analogous situation in the non-orien... | 37 | https://mathoverflow.net/users/1822 | 20759 | 13,779 |
https://mathoverflow.net/questions/20698 | 6 | I've come up with the following optimization problem in my research. Is this a known problem in graph-theory and/or combinatorial optimization? If not, which of the known problems are the most similar to it?
Let's have a graph $G=(V,E)$ with real positive or negative weights assigned to its edges: $w: E \rightarrow ... | https://mathoverflow.net/users/5223 | Degree constrained edge partitioning | I'm not sure about published references to this specific problem, but I'm pretty sure it can be solved in polynomial time via a reduction to minimum weight perfect matching, as follows.
Replace each vertex of degree $d$ by a complete bipartite graph $K\_{d-2,d}$ where each of the original edges incident to the vertex... | 3 | https://mathoverflow.net/users/440 | 20763 | 13,783 |
https://mathoverflow.net/questions/20773 | 8 | The above title is in fact a special case of what I want to ask.
Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}\_p $. The intersections of these extensions to the ring of integer of the absolute algebraic closure of $\mathbb{Q}\_p$ give us a notion of Galois extensions for $\mathbb{Z}\_... | https://mathoverflow.net/users/2701 | Is there a notion of Galois extension for Z / p^2? | Perhaps not directly answering your questions but something along those lines is Deligne's theory of truncated valuation rings, given in [Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique 0](http://www.ams.org/mathscinet-getitem?mr=771673).
A truncated valuation ring is an Artin loca... | 4 | https://mathoverflow.net/users/3143 | 20783 | 13,792 |
https://mathoverflow.net/questions/20782 | 50 | ### Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. Furthermore, we know that Spec$(A)$ has a base of open affine subsets, the "basic" or "principal" open affines $D(f)$ ... | https://mathoverflow.net/users/778 | Ring-theoretic characterization of open affines? | **Theorem 1**: Let $R$ be an integral domain with field of fractions $K$, and $R \to A$ a homomorphism. Then $Spec(A) \to Spec(R)$ is an open immersion if and only if $A=0$ or $R \to K$ factors through $R \to A$ (i.e. $A$ is birational over $R$) and $A$ is flat and of finite type over $R$.
Proof: Assume $Spec(A) \to ... | 40 | https://mathoverflow.net/users/2841 | 20792 | 13,797 |
https://mathoverflow.net/questions/20789 | 10 | Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ up to the $m^{\rm th}$ order:
(1) $\mathbb{E}[X^k] = \mathbb{E}[Z^k]$ for $k = 1, \ldots m$.
Here is a positive resu... | https://mathoverflow.net/users/3736 | approximate a probability distribution by moment matching | For (1) and (2) just forget about probability and recall everything you ever learned about orthogonal polynomials and the Gauss quadrature formulae.
3) is false as stated: there are plenty of Schwartz functions orthogonal to all polynomials, so you can have all moments coincide and still have a large distance (in any... | 5 | https://mathoverflow.net/users/1131 | 20793 | 13,798 |
https://mathoverflow.net/questions/20788 | 2 | Mathematically, I know what a semigroup is: It is a set S along with an *associative* binary operation $\* : S \times S \rightarrow S$. So far, so good.
From a computational perspective, one can represent a semigroup as the tuple $\left< S,\* \right>$, or my preference, as a record { S: type; $\* : S \times S \righta... | https://mathoverflow.net/users/3993 | What is a semigroup or, what do I do with that associativity proof? | The answer to your real question is that all proofs are typically equated in set theories (more accurately, in topoi). That is, the interpretation of a proposition is as an element of the lattice of truth values, which in classical logic can take on either the value true or false.So, this means that if you give a semig... | 3 | https://mathoverflow.net/users/1610 | 20799 | 13,801 |
https://mathoverflow.net/questions/20764 | 13 | This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first cohomology $H^1(\Sigma,U(1))$, where $U(1)$ is just complex numbers of norm 1, parametrizes homomorphisms $$\pi\_{1}(\... | https://mathoverflow.net/users/1622 | Representations of \pi_1, G-bundles, Classifying Spaces | Since it is not always clear what $H^1(X;G)$ means for a non-abelian group, Atiyah might have meant this loosely, perhaps using Cech 1-cocycles to construct flat $G$ bundles, with homologous cycles giving isomorphic bundles. The moduli space of flat bundles is homeomorphic (and real analytically isomorphic) to the spac... | 6 | https://mathoverflow.net/users/3874 | 20804 | 13,804 |
https://mathoverflow.net/questions/20802 | 7 | Let $f: X\to Y$ be a finite (surjective) morphism between two algebraic varieties. I know when $X$ and $Y$ are non-singular and $\dim Y =1$, $f$ is flat. But in general, is it true that $f$ is a flat morphism?
| https://mathoverflow.net/users/2348 | Finite morphisms between algebraic varieties are flat? | If $X$ and $Y$ are both regular, then this is true. In fact, it's true more generally if $Y$ is regular and $X$ is Cohen-Macaulay (Eisenbud, Commutative Algebra, Corollary 18.17). In general it's certainly false.
| 15 | https://mathoverflow.net/users/1594 | 20806 | 13,806 |
https://mathoverflow.net/questions/20225 | 13 | I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like iid random variables.
I am also aware of some facts (like inequalities for square functions vs. Khinchine Inequality) ... | https://mathoverflow.net/users/5112 | Why do Littlewood-Paley projections behave like iid random variables | It is much better to replace 'iid random variables' above by 'martingale differences.'
The usual Littlewood-Paley square function is closely related to the Haar square function.
And the Haar square function is exactly a martingale square function, namely a sum of
squares of martingale differences.
One can pas... | 7 | https://mathoverflow.net/users/1158 | 20808 | 13,807 |
https://mathoverflow.net/questions/20791 | 12 | Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are isogeneous. But do we actually have $B \cong D$?
| https://mathoverflow.net/users/5197 | isomorphism of abelian varieties | This is false even for elliptic curves over $\mathbb{C}$. This was proved by T. Shioda in "Some remarks on abelian varieties" J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 11-21, <http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6164/1/jfs240102.pdf>.
| 20 | https://mathoverflow.net/users/4790 | 20811 | 13,810 |
https://mathoverflow.net/questions/20784 | 4 | Hi everyone,
Let $X$ be a smooth projective variety over a field $k$ and let $L$ be a line bundle on $X$. I'm reading the article *Heights for line bundles on arithmetic varieties* and there one speaks of $\textrm{Pic}^L(X)$. What is that? And above that, if $L$ and $K$ are algebraically equivalent line bundles, why ... | https://mathoverflow.net/users/4333 | For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X) | Let me address the last part of your question.
Let $X$ be a smooth, projective variety over an arbitrary ground field $k$.
I want to write $Pic^{[L]}(X)$ instead of $Pic^L(X)$ -- i.e., to make explicit that the variety depends only on the Neron-Severi class of $L$ -- for reasons which will become clear shortly.
S... | 4 | https://mathoverflow.net/users/1149 | 20813 | 13,812 |
https://mathoverflow.net/questions/20777 | 11 | A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: each point (equivalently, some point) is an isolated fixed point of some (holomorphic, isometric) involution of the space... | https://mathoverflow.net/users/379 | Hermitian symmetric spaces vs Hermitian homogeneous spaces | Here is a geometric answer to (2), or more precisely a slight and classical reinterpretation. First, note that Hermitian symmetric spaces fit into the more general concept of riemannian symmetric space; this can helps you find references.
Denote your space by $X$ (assumed to be Riemannian, or Hermitian if you prefer)... | 5 | https://mathoverflow.net/users/4961 | 20819 | 13,814 |
https://mathoverflow.net/questions/20740 | 227 | The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of Kolmogorov, i.e., probability measures) are appealing and potentially useful to me. It seems to me that, perhaps more than ... | https://mathoverflow.net/users/1149 | Is there an introduction to probability theory from a structuralist/categorical perspective? | $\def\Spec{\mathop{\rm Spec}}
\def\R{{\bf R}}
\def\Ep{{\rm E}^+}
\def\L{{\rm L}}
\def\EpL{\Ep\L}$
One can argue that an object of the right category of spaces in measure theory is not a set equipped with a σ-algebra of measurable sets,
but rather a set $S$ equipped with a σ-algebra $M$ of measurable sets and a σ-ideal ... | 231 | https://mathoverflow.net/users/402 | 20820 | 13,815 |
https://mathoverflow.net/questions/20827 | 25 | A group $G$ is [Hopfian](http://en.wikipedia.org/wiki/Hopfian_group) if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth manifolds Hopfian?
Perhaps the manifold structure is irrelevant and makes... | https://mathoverflow.net/users/4354 | Are fundamental groups of aspherical manifolds Hopfian? | The Baumslag-Solitar group $B(2,3)=\langle a,b\vert ba^2b^{-1}=a^3\rangle$ is not Hopfian. But it has a natural $K(\pi,1)$ given by the double mapping cylinder of $S^1 \rightrightarrows S^1$ where the maps are $z\mapsto z^2$ and $z\mapsto z^3$. This is a finite CW complex.
---
Edit: The double mapping cylinder ca... | 19 | https://mathoverflow.net/users/250 | 20829 | 13,820 |
https://mathoverflow.net/questions/20826 | 13 | It is straightforward to show, for example, that the set of zero divisors of a (commutative unital) reduced Noetherian ring is precisely the union of its minimal primes. When else can we say that the set of zero divisors is equal to the union of the minimal primes? Are there other useful cases where this is true? Is th... | https://mathoverflow.net/users/1353 | When is the set of zero divisors equal to the union of the minimal primes in a reduced ring? | The answer is: *always*, and argument is pretty simple:
Let $R$ be a reduced commutative unital ring.
If $a\in R$ is a zero divisor, then $ab=0,$ for some $b\neq 0.$ Hence
$b\not \in 0 = \text{nil}(R)= \bigcap \text{Spec}(R) =\bigcap \text{Specmin}(R),$ where $\text{Specmin}(R)$ states for family of all minimal prime... | 23 | https://mathoverflow.net/users/5080 | 20833 | 13,823 |
https://mathoverflow.net/questions/20671 | 36 | This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant singular cohomology
$H^{\ast}\_G(G) = H^{\ast}( (G\times EG)/G )$, with integer coefficients, say.
>
> What is $H... | https://mathoverflow.net/users/2356 | What is the equivariant cohomology of a group acting on itself by conjugation? | I asked Dan Freed, who gave a very clean general solution to this problem (as expected).
Here it is (all mistakes in the transcription are mine of course).
The claim is that the equivariant cohomology of G acting on G is indeed the tensor product of cohomology of BG with cohomology of G - in other words the Leray spe... | 42 | https://mathoverflow.net/users/582 | 20849 | 13,831 |
https://mathoverflow.net/questions/15074 | 2 | Assume $M$ is an open 3-manifold which can be deformation retracted to a point. Is it necessarily homeomorphic to $\mathbb R^3$?
(I know Whitehead had an example which is contractible and not homeomorphic to $\mathbb R^3$
Does his counterexample strong deformation retract to a point?)
| https://mathoverflow.net/users/3922 | Must a Strong deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3$? | Allen Hatcher's comment is actually an answer: "Just to clarify: The original question seems to be about the distinction between a space being contractible and the possibly stronger condition of deformation retracting to a point. For nice spaces (manifolds, CW complexes, ...) the two conditions are in fact equivalent. ... | 4 | https://mathoverflow.net/users/66 | 20852 | 13,833 |
https://mathoverflow.net/questions/20850 | 2 | Let $f : X \to Y$ be a morphism of schemes. Is it possible to associate to every closed immersion $i : F \to X$ a closed immersion $f^\* i : G \to Y$, such that in the affine case, $(Spec(A) \to Spec(B))^\*$ is given taking ideals of $A$ to ideals of $B$ via preimages? Probably this is well-known to every algebraic geo... | https://mathoverflow.net/users/2841 | is there a push-forward of closed subschemes? | It sounds as though what you want is the closure of the image of $F$ under $f$. (That is, the minimal closed subscheme of $Y$ factoring $f$.)
If $X =$ Spec $A$, and $Y =$ Spec $B$, and $F =$ Spec $A/I$, and $f$ corresponds to the ring map $f':B\to A$, then we can consider the preimage $J$ of $I$ under $i'$. Consider ... | 6 | https://mathoverflow.net/users/5254 | 20854 | 13,835 |
https://mathoverflow.net/questions/20831 | 20 | It is known (given CFSG) that all non-abelian finite simple groups have small outer automorphism groups. However, it's quite tedious to list all the possibilities. Does anyone know a reference for a statement of the following form?
Let G be a non-abelian finite simple group. Then |Out(G)| < f(|G|) (where f is somethi... | https://mathoverflow.net/users/4053 | Estimate for the order of the outer automorphism group of a finite simple group | Check article "Probabilistic generation of wreath products of non-abelian finite simple groups" by Martyn Quick. In Section 3.1 he consider this question and get $|Out G|\leq |G|/30$ for every non-abelian finite simple group $G$, which was enough for his needs.
| 15 | https://mathoverflow.net/users/4408 | 20859 | 13,838 |
https://mathoverflow.net/questions/20856 | 10 | Let $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n^+}$ be a triangular decomposition of semisimple Lie algebra. Let $\mathcal{Z}$ be the central of universal envoloping Lie algebra of $\mathfrak{g}$.
Let $\mathcal{C}$ be the category of representations of $\mathfrak{g}$, on which $\mathfrak{n ^+}$ an... | https://mathoverflow.net/users/5082 | Twin categories in representation of Lie algebra | The equivalence (or something very close, I haven't checked carefully what is written) follows from Beilinson-Bernstein localization. The two categories can be realized roughly speaking as D-modules on B\G/N and N\G/B, and the equivalence comes from the interchange of the two sides. Slightly more precisely, Beilinson-B... | 8 | https://mathoverflow.net/users/582 | 20862 | 13,841 |
https://mathoverflow.net/questions/20863 | 6 | First some simple observations in order to motivate the question:
The functor $Set^{op} \to Set, X \to \{\text{subsets of }X\}, f \to (U \to f^{-1}(U))$ is representable. The representing object is $\{0,1\}$ with the universal subset $\{0\}$. Also the functor $Top^{op} \to Set, X \to \{\text{open subsets of }X\}$ is ... | https://mathoverflow.net/users/2841 | Is the functor of open subschemes representable? | No, it is not representable. If it were, the functor of open subset for schemes over a field $k$ would also be representable by the base change to $\mathop{\rm Spec} k$ of the representing object. Suppose that this last functor is represented by an open embedding of $k$-schemes $S\_1 \subseteq S$. If $T$ is a $k$-schem... | 14 | https://mathoverflow.net/users/4790 | 20866 | 13,842 |
https://mathoverflow.net/questions/20399 | 17 | What is the [conjugate prior](http://en.wikipedia.org/wiki/Conjugate_prior) distribution of the Dirichlet distribution?
---
Edit: Since I asked this question many years ago, I've written a [Python library](https://github.com/NeilGirdhar/efax) for working with exponential families. [Maximum likelihood estimation o... | https://mathoverflow.net/users/634 | Conjugate prior of the Dirichlet distribution? | [Neil](https://mathoverflow.net/users/634/neil) sent me an email asking:
===
I read your post at <http://www.stat.columbia.edu/~cook/movabletype/archives/2009/04/conjugate_prior.html> and I was wondering if you could expand on how to update the Dirichlet conjugate prior that you provided in your [paper](http://cvsp... | 22 | https://mathoverflow.net/users/5258 | 20875 | 13,846 |
https://mathoverflow.net/questions/20874 | 9 | Probably a well-know question, but I haven't solved it, so I'll ask.
I can show that every matrix in $M\_2(\mathbb{R})$ is the sum of two squares of matrices in $M\_2(\mathbb{R})$.
If $n>2$, I can also show that every matrix in $M\_n(\mathbb{R})$ is the sum of three squares of matrices in $M\_n(\mathbb{R})$.
So my ... | https://mathoverflow.net/users/3958 | Waring's problem for matrices | The answer is YES if $n$ is even. But if $n$ is odd, then the answer is NO since $-I$ is not a sum of two squares.
See
Griffin and Krusemeyer, Matrices as sums of squares, *Linear and Multilinear Algebra* **5** (1977/78), no. 1, 33-44
for the proofs of these facts and generalizations.
| 14 | https://mathoverflow.net/users/2757 | 20876 | 13,847 |
https://mathoverflow.net/questions/20147 | 4 | It's all in the title basically. There's an interesting topic called [systolic geometry](http://en.wikipedia.org/wiki/Systolic_geometry) that has grown a lot in the past 30 years, with a (first?) [textbook](http://www.ams.org/bookstore-getitem/item=surv-137) on the subject by M.Katz (AMS 2007).
So I was wondering wha... | https://mathoverflow.net/users/469 | What would a graduate course on systolic geometry typically cover? | This is interesting. I imagine that any course would vary quite a bit depending on who taught it.
Any course should probably contain Gromov's proof of the systolic inequality for essential manifolds. Other than that, I am not sure. The course could dive into systoles on surfaces and some of the arithmetic constructio... | 4 | https://mathoverflow.net/users/5261 | 20878 | 13,848 |
https://mathoverflow.net/questions/15176 | 9 | A sequence $a\_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a\_{mn} = a\_m a\_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum\_n a\_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in addition $\limsup \frac{\log \lvert a\_1 + \ldots + a\_n \rvert}{\log n} = 0$, then the series is (conditionally) con... | https://mathoverflow.net/users/3755 | Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1 | As a matter of fact, it isn't hard to construct a multiplicative sequence $a\_n$ such that $f(z)$ is an entire function without zeroes. Unfortunately, it is completely useless for the questions that you brought up as "motivation".
Here is the construction.
Claim 1: Let $\lambda\_j\in [0,1]$ ($j=0,\dots,M$). Assume ... | 8 | https://mathoverflow.net/users/1131 | 20888 | 13,855 |
https://mathoverflow.net/questions/20838 | 11 | My question is very basic (I don't know too much of differential geometry):
given a fiber bundle, is there a necessary and sufficient condition for its tangent bundle to be trivial?
I have some ideas, but submitted to some conditions on the cohomology ring of the bundle.
(I apologize if it is trivial.)
| https://mathoverflow.net/users/4770 | Is there a necessary and sufficient condition for the tangent bundle of a fiber bundle to be trivial? | This is far from being a complete answer, but there is a case when one construct a parallelizable bundle (meaning its total space has trivial tangent bundle) from a given (geometric) bundle.
The context is that of $G$-structure, which are a formalization of the concept of geometric structures. A $G$-structure is a s... | 10 | https://mathoverflow.net/users/4961 | 20904 | 13,863 |
https://mathoverflow.net/questions/20913 | 3 | Given an arithmetic variety $f: X \rightarrow Spec(\mathbb{Z})$.
Is there a notion of boundedness for families of sheaves on $X$?
I only found the notion for families on the fibers of $f$. But i am interested in sheaves defined on $X$.
All definitions / theorems i found only work when $X$ is defined over some fie... | https://mathoverflow.net/users/3233 | Families of sheaves on arithmetic varieties | When doing moduli theory over $\mathbb Z$, or another base scheme, one works with sheaves that are flat over the base; this implies that all the discrete invariants, such as the Hilbert polynomial, are constant in the fibers. Stability is defined fiber by fiber; i.e., a sheaf is (semi)stable when it it (semi)stable on ... | 3 | https://mathoverflow.net/users/4790 | 20917 | 13,872 |
https://mathoverflow.net/questions/20891 | 13 | Let $M$ be a differentiable manifold, $\Delta$ the closed simplex $[p\_0, p\_1,...,p\_k]$. A differential singular $k$-simplex $\sigma$ of $M$ is a smooth mapping $\sigma:\Delta \to M$.
And we construct a chain complex in the same way we construct the chain complex of singular homology, we gain its homology group.
... | https://mathoverflow.net/users/4621 | singular homology of a differential manifold | This depends on what exactly is a smooth mapping from a simplex to the manifold. The standard definition is that the mapping of a non-open subset $X$ of $\mathbf{R}^n$ to a manifold is smooth iff it can be extended to a smooth mapping of an open neighborhood of $X$. With this definition the comparison theorem is true a... | 16 | https://mathoverflow.net/users/2349 | 20926 | 13,876 |
https://mathoverflow.net/questions/20925 | 3 | Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?
| https://mathoverflow.net/users/nan | finite generation of the Mordell-Weil group over finitely generated fields | It is in Lang's book "Fundamentals of diophantine geometry", chapter 6:
[google book preview](http://books.google.co.uk/books?id=ShRel9iqsmUC&lpg=PA138&dq=lang%2520serge%2520mordell-weil%2520finitely&pg=PA138#v=onepage&q&f=false)
| 3 | https://mathoverflow.net/users/5015 | 20928 | 13,877 |
https://mathoverflow.net/questions/20929 | 18 | This question is something of a follow-up to
[Transformation formulae for classical theta functions](https://mathoverflow.net/questions/19400/) .
How does one recognise whether a subgroup of the modular group
$\Gamma=\mathrm{SL}\_2(\mathbb{Z})$ is a congruence subgroup?
Now that's too broad a question for me to exp... | https://mathoverflow.net/users/4213 | Distinguishing congruence subgroups of the modular group | There is one answer in the following paper, along with a nice bibliography of other techniques:
[MR1343700 (96k:20100)](http://www.ams.org/mathscinet-getitem?mr=1343700)
Hsu, Tim(1-PRIN)
Identifying congruence subgroups of the modular group. (English summary)
Proc. Amer. Math. Soc. 124 (1996), no. 5, 1351--1359.
| 19 | https://mathoverflow.net/users/317 | 20931 | 13,879 |
https://mathoverflow.net/questions/20927 | 8 | Let E be an elliptic curve over the rationals with conductor $Mp^2$ with p>5 and M and p coprime, and let $\rho$ be the Galois representation attached to the p-torsion points of E. Is there a way to consider deformations of $\rho$ to representations that still have "additive reduction at p" with Serre weight 2? Actuall... | https://mathoverflow.net/users/92 | Is there an R=T type result for modular forms with additive reduction? | If you fix a prime $\ell$, and consider the Galois action of the decomposition group $D\_p$ on the (rational) $\ell$-adic Tate module (here "rational" means tensored with $\mathbb Q\_{\ell}$),
then (in a standard way, due to Deligne) you can convert this action into a representation
of the Weil--Deligne group, and so i... | 7 | https://mathoverflow.net/users/2874 | 20932 | 13,880 |
https://mathoverflow.net/questions/20922 | 7 | Tonight, a friend of mine give me a concise introduction to Shimura variety . I only get some first impression of it. I think the hodge structure is a generalization of the cohomology ring of Kaehler manifold or algebraic manifold , and i think of the Shimura variety an anologue of the analytic familly of complex manif... | https://mathoverflow.net/users/4437 | what is the motivation of Shimura variety? | The theory of Shimura varieties was begun by Shimura, and further developed by Langlands (who introduced the name), and is now a central part of arithemtic geometry and of the theories
of automorphic forms, Galois representations, and motives.
Shimura varieties are certain moduli of Hodge structures; but that is perh... | 21 | https://mathoverflow.net/users/2874 | 20950 | 13,892 |
https://mathoverflow.net/questions/18460 | 9 | Sorry for my poor English.
Let $X$ be a reducible projective variety.
**My question is:**
1. How can I compute the dualizing sheaf of $X$ and express it in an explicit way?
2. Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible comp... | https://mathoverflow.net/users/4643 | Dualizing sheaf of reducible variety? | With regards part 2.
Let's assume that you have two components $X\_1$ and $X\_2$ (or even unions of components) such that $X\_1 \cup X\_2 = X$=. Let $I\_1$ and $I\_2$ denote the ideal sheaves of $X\_1$ and $X\_2$ in $X$.
Set $Z$ to be the *scheme* $X\_1 \cap X\_2$, in other words, the ideal sheaf of $Z$ is $I\_1 +... | 14 | https://mathoverflow.net/users/3521 | 20954 | 13,896 |
https://mathoverflow.net/questions/20955 | 6 | It is known that if GRH holds there does not exist additional Idoneal numbers. (see www.mast.queensu.ca/~kani/papers/idoneal.pdf this paper puts on the question of correctnes for Wikipedia and Wolfram MathWorld since they state there could only be ONE additional Idoneal number)
What I am interested in is whether ther... | https://mathoverflow.net/users/1737 | The missing Euler Idoneal numbers | If X is an idoneal number, then the class group of discriminant -4X has exponent dividing 2, so the class number is equal to the number of genera (Theorem 6 in Kani's paper: for me, this is the most convenient definition), which is given by an explicit recipe in terms of the number of prime factors of X and its congrue... | 6 | https://mathoverflow.net/users/1149 | 20957 | 13,897 |
https://mathoverflow.net/questions/20960 | 155 | I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial directly, but the integral definition $\Gamma(z) = \int\_0^\infty t^{z-1} e^{-t}\,dt$, makes more sense as $\Pi(z) = \int\_... | https://mathoverflow.net/users/5279 | Why is the Gamma function shifted from the factorial by 1? | From Riemann's Zeta Function, by H. M. Edwards, available as a Dover paperback, footnote on page 8: "Unfortunately, Legendre subsequently introduced the notation $\Gamma(s)$ for $\Pi(s-1).$Legendre's reasons for considering $(n-1)!$ instead of $n!$ are obscure (perhaps he felt it was more natural to have the first pole... | 127 | https://mathoverflow.net/users/3324 | 20962 | 13,899 |
https://mathoverflow.net/questions/20924 | 8 | Let $A, B\in M\_{n}(\mathbb{R})$ be symmetric positive definite matrices. It is easy to see $Tr(A^2+AB^2A)=Tr(A^2+BA^2B)$. Numerical experiments indicate $$Tr[(A^2+AB^2A)^{-1}]\ge Tr[(A^2+BA^2B)^{-1}],~~(1)$$ but it seems difficult to show it.
Remark. When $n=2,3$, by direct computation, (1) is true. Here is an expr... | https://mathoverflow.net/users/3818 | A question on a trace inequality | Note first that $A^2+AB^2A=(A+iAB)(A-iBA)$. The reverse product is $(A-iBA)(A+iAB)=A^2+BA^2B-i(BA^2-A^2B)=X-iC$. Thus, the quantity on the left is $\operatorname{Tr} (X-iC)^{-1}$ and that on the right is $\operatorname{Tr} X^{-1}$. Moreover, the self-adjoint complex matrix $X-iC$ is positive definite (as the product of... | 19 | https://mathoverflow.net/users/1131 | 20975 | 13,906 |
https://mathoverflow.net/questions/20978 | 3 | Let $X$ be a quasi-compact scheme and let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}\_X$-algebras on $X$. $X$ being quasi-compact, we can write $X = U\_1 \cup \dots \cup U\_n$ with each $U\_i$ affine, say $U\_i =\textrm{Spec}R\_i$, and such that $\mathcal{A}\mid\_{U\_i} \simeq \widetilde{A\_i}$ for some fi... | https://mathoverflow.net/users/493 | Quasi-coherent sheaves of O_X-algebras | Under some slightly stronger hypothesis (Noetherian is certainly enough) we may write
$\mathcal A$ as the union of its coherent subsheaves. If $\mathcal E$ is a coherent subsheaf, then the subalgebra of $\mathcal A$ that it generates will also be coherent,
because this can be tested locally, where it then follows from ... | 3 | https://mathoverflow.net/users/2874 | 20983 | 13,909 |
https://mathoverflow.net/questions/20298 | 10 | This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x\_1$, $x\_2$, ..., $x\_n$ be $n$ reals. For any integer $k$, define a real $f\_k\left(x\_1,x\_2,...,x\_n\right)$ as the sum
$\sum\limits\_{T\subseteq\left\lbrace 1,2,...,n\right\rb... | https://mathoverflow.net/users/2530 | Sum of difference moduli vs. sum of modulus differences | Hi, Darij!
This is actually quite simple (and also much more appropriate for AoPS than for MO). The idea is to show that for every $t$, the expression $\sum\_T|t+D\_T|$, where $D\_T$ is your difference, goes down if you replace all $x\_k$ by their absolute values ($t=0$ is your claim). The base $n=2$ is rather trivia... | 5 | https://mathoverflow.net/users/1131 | 20985 | 13,911 |
https://mathoverflow.net/questions/20996 | 4 | I would like to know under what condition the morphism $\mathcal{O}\_Y\longrightarrow f\_\ast \mathcal{O}\_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective.
Let me give an example (which I'm not completely sure about though).
I believe, if $X$ and $Y$ are reduced and $f$ is surjective and clo... | https://mathoverflow.net/users/4333 | Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective | If $f$ is quasi-compact and quasi-separated then the kernel of the map $O\_Y\to f\_\*(O\_X)$ consists of locally nilpotent elements if and only if $f(X)\subset Y$ is a dense set.
| 6 | https://mathoverflow.net/users/781 | 21000 | 13,919 |
https://mathoverflow.net/questions/20997 | 3 | I was just going to check something in FGA and I didn't have access to my pdf-copy, so I did what I normally do when in such a circumstance: surf to the Grothendieck circle's webpage.
And what did I find there? All mathematical texts written by Grothendieck himself was removed "per his request".
Does anyone know ... | https://mathoverflow.net/users/2147 | What's up with the Grothendieck circle? | Grothendieck seems to have requested that all his materials be taken town. As pointed out in comments, this has been discussed fairly [thoroughly](http://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter/) on [the blogs](http://golem.ph.utexas.edu/category/2010/02/grothendieck_said_stop.html) and, in fact, [on me... | 5 | https://mathoverflow.net/users/66 | 21005 | 13,922 |
https://mathoverflow.net/questions/20968 | 1 | Hi, my apologies for a rather non-specific question. I wonder if there is a general set of conditions under which operators are commutative in functional analysis. Most that I've found is that "operators are, in general, not commutative". Is there any reference someone could point me to for some kind of review or speci... | https://mathoverflow.net/users/5282 | rules for operator commutativity? | One obvious but important observation is that, for operators on a $n$-dimensional vector space over a field, if $1 < n < \infty$, we have $AB \neq BA$ *generically*. In other words, consider the commutativity locus $\mathcal{C}\_n$ of all pairs of $n \times n$ matrices $A,B$ such that $AB = BA$ as a subset of $\mathbb{... | 8 | https://mathoverflow.net/users/1149 | 21007 | 13,924 |
https://mathoverflow.net/questions/20963 | 6 | It is known that Namba forcing is stationary-preserving and hence can be used in the setting of Martin's Maximum. Does this result in any striking consequences?
| https://mathoverflow.net/users/4706 | What are the Martin's Maximum consequences of Namba forcing? | I think that I may have found a suitable candidate; namely, the result of
Konig and Yoshinobu that $MM$ implies that there are no $\omega\_{1}$-regressive $\omega\_{2}$-Kurepa trees. The proof seems to have the same relatively direct flavor as those in Baumgartner's $PFA$ article.
| 5 | https://mathoverflow.net/users/4706 | 21008 | 13,925 |
https://mathoverflow.net/questions/20600 | 2 | $X$ = bi-elliptic surface (smooth and over $\mathbb{C}$),
Aut($X$) = the group of automorphisms of $X$,
Aut$^0(X)$ = connected component of the identity in Aut($X$).
Is Aut$^0(X)$ always an affine algebraic group?
| https://mathoverflow.net/users/5197 | Automorphism group of bi-elliptic surface | The answer is always "no". By classification, a bielliptic surface over $\mathbb C$ has the form $(E\times F)/G$ where $E,F$ are elliptic curves, $G=\subset Aut(E,0)$ is an abelian group acting by complex multiplications on $E$ and by translations on $F$. ($G$ is not necessarily cyclic as Tuan correctly points out.)
... | 4 | https://mathoverflow.net/users/1784 | 21009 | 13,926 |
https://mathoverflow.net/questions/20867 | 6 | Hi all,
given (a1,...,an) formed by distinct letters, it's a well known problem to count the number of permutations with no fixed element.
I've been trying to solve a generalization of this problem, when we allow repetition of the letters.
I was able only to partially solve the problem when we have only repetitio... | https://mathoverflow.net/users/1172 | Derangements with repetition | The formula based on Inclusion-Exclusion for the usual number $D(n)$ of derangements of $n$ objects can be generalized. The result is the following.
Fix $k\geq 1$. Let $\mathbb{N}=\lbrace 0,1,2,\dots\rbrace$. For $\alpha=(\alpha\_1,\dots,\alpha\_k)\in\mathbb{N}^k$, let $D(\alpha)$ be the number of fixed-point free pe... | 16 | https://mathoverflow.net/users/2807 | 21019 | 13,928 |
https://mathoverflow.net/questions/21018 | 2 | Let $f:X\longrightarrow \mathbf{P}^1$ be a finite morphism of schemes. If $f$ is etale and $X$ is connected, we can show that $f$ is an isomorphism. That is, the projective line is simply connected. I would like to know if something more general can be said about $f$ if it is not assumed to be etale.
In this general... | https://mathoverflow.net/users/4333 | What is known about finite morphisms from X to the projective line | No because $f$ can be ramified pretty much anywhere. Just think of a random rational function $f=p(x)/q(x)$ with $p,q$ coprime polynomials, $p$ non-constant and $q$ non-zero. That gives a finite morphism from the projective line to itself that is in general much more complicated than the map you suggest.
| 6 | https://mathoverflow.net/users/1384 | 21020 | 13,929 |
https://mathoverflow.net/questions/15645 | 4 | I wanted recently to discuss with a fairly elementary mathematics class the kinds of self-maps of Euclidean space that carry triangles to triangles. Obviously linear maps do this, and it seemed just as obvious to me that they were the only ones (up to translation).
I asked a colleague, and he pointed out an obvious c... | https://mathoverflow.net/users/2383 | Non-affine, projective vector field on $R^n$ | I don't know what was meant in that exercise, but your revised conjecture is certainly true and well-known. Here is an elementary proof.
The assumptions (local injectivity, continuity and segment-to-segment mapping) imply that the map is injective and preserves collinearity of points and endpoints of segments. By con... | 7 | https://mathoverflow.net/users/4354 | 21022 | 13,930 |
https://mathoverflow.net/questions/10603 | 21 | Let $E\_1$ and $E\_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-function of $E\_1$ and $E\_2$ are same . Does it mean that the complex $L$-function of $E\_1$ and $E\_2$ are also same ?
Is... | https://mathoverflow.net/users/2876 | Does p-adic $L$- function determine the $L$ function | Hmmm. I am not so sure that the answer is *"No"*. In fact I would rather bet on *"Yes"*.
Of course, I totally agree that the characteristic ideal, i.e. the ideal generated by the $p$-adic $L$-function in $\Lambda$, is not enough to determine the elliptic curve. In particular there are plenty of curves for which the $... | 13 | https://mathoverflow.net/users/5015 | 21029 | 13,933 |
https://mathoverflow.net/questions/18833 | 10 | Is there some generalization of the Jordan-Hölder decomposition for group objects in a category $\mathcal{C}$?
If $\mathcal{C}$ is the category Sch$(S)$ of schemes over a base scheme $S$ then (I think) this is true, also probably for other categories of "spaces" like Top or Diff it should be true, but I don't have an... | https://mathoverflow.net/users/4619 | Jordan Hölder decomposition for group objects | You can start by looking at the paper by P.J. Hilton and W. Ledermann, "On the Jordan-Hölder theorem in homological monoids", where three axioms are needed to establish the decomposition, and the third one is essentially guaranteeing the second isomorphism theorem.
In "Mal'cev, protomodular, homological and semi-abel... | 6 | https://mathoverflow.net/users/2384 | 21032 | 13,935 |
https://mathoverflow.net/questions/20705 | 6 | As per [a recent question of mine](https://mathoverflow.net/questions/20228/omega-1-times-beta-mathbbn-normal), $\omega\_1 \times \beta \mathbb{N}$ is not normal. I'm wondering whether there's some sort of "natural" condition that describes when a space has a normal product with $\beta \mathbb{N}$, analagous to Dowker'... | https://mathoverflow.net/users/4959 | Is there a "natural" characterization of when X × βN is normal? | The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita ([*Paracompactness and product spaces*](http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50119.pdf), MR[132525](http://www.ams.org/mathscinet-getitem?mr=132525)), where he generalizes ... | 6 | https://mathoverflow.net/users/2000 | 21033 | 13,936 |
https://mathoverflow.net/questions/21031 | 14 | This question was actually asked by John Stillwell in a comment to an answer to this [question](https://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice). I thought I would advertise it as a separate question since no one has yet answered and I am also curious about it.
**Question:... | https://mathoverflow.net/users/2233 | Ultrafilters vs Well-orderings | It is consistent that there exists a non-principal ultrafilter over $\omega$ while $\mathbb{R}$ is not well-ordered. To see this, suppose that the
partition relation $\omega \to (\omega)^{\omega}$ holds in $L(\mathbb{R})$.
Then forcing with $\mathbb{P}= [\omega]^{\omega}$ adjoins a selective ultrafilter $\mathcal{U}$ o... | 15 | https://mathoverflow.net/users/4706 | 21034 | 13,937 |
https://mathoverflow.net/questions/21028 | 3 | Apologies for the vague title - I couldn't come up with a single sentence that summarised this problem well. If you can, please edit or suggest a better one!
This question is also rather specific and contains lots of annoying technical detail. I must admit to not really expecting an answer unless there's an obvious s... | https://mathoverflow.net/users/4959 | Shape of long sequences in C(ω_1) | I claim that there are no bad sequences in C(ω1).
Suppose to the contrary that xα is bad. For any countable
ordinal β, there is rβ in
C(ω1) such that the distance between rβ and
xα for α < β is at most 1.
For any countable ordinal β and any positive
rational number ε, there is a smaller ordinal
γ < β such that all
rβ... | 3 | https://mathoverflow.net/users/1946 | 21055 | 13,953 |
https://mathoverflow.net/questions/21051 | 14 | Illusie's article about étale cohomology [available here](http://www.math.u-psud.fr/~illusie/Grothendieck_etale.pdf) (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not give the right answers in the case of étale cohomology.
He s... | https://mathoverflow.net/users/362 | Why does the naive definition of compactly supported étale cohomology give the wrong answer? | It is important in etale cohomology, as it is topology, to define cohomology
groups with compact support --- we saw this already in the case of curves in
Section 14. They should be dual to the ordinary cohomology groups.
The traditional definition (Greenberg 1967, p162) is that, for a manifold
$U$,
$
H\_{c}^{r}(U,\ma... | 15 | https://mathoverflow.net/users/930 | 21056 | 13,954 |
https://mathoverflow.net/questions/21062 | 6 | I have never understood the trace map,not even after reading [Geometric Interpretation of Trace](https://mathoverflow.net/questions/13526/geometric-interpretation-of-trace). The problem with many answers in the above discussion is the geometric intuition does not apply to other field.
As I don't want this to be clos... | https://mathoverflow.net/users/2701 | Is there good intution of the trace map? | I don't know if this is what you're looking for, but there's a basis-free definition of the trace in general, outside of the algebraic number theory context -- a linear transformation $V\to V$ corresponds in a natural way to an element of $V\otimes V^\ast$, and the trace map is the map $V\otimes V^\ast\to k$ induced by... | 10 | https://mathoverflow.net/users/5281 | 21064 | 13,960 |
https://mathoverflow.net/questions/21014 | 2 | I found the following paragraph in the paper " Intro to symplectic field theory "
which I don't understand what does it mean precisely?
Suppose W is a symplectic (or Kahler) manifold.
D, smooth divisor in it.
then $\tilde{W}= W-D $ is a Weinstein manifold which contains an isotropic deformation retract $\Delta$.
... | https://mathoverflow.net/users/5259 | isotropic deformation retract of Weinstein manifolds? | This story begins with the Lefschetz hyperplane theorem - the fact that if $D$ is a smooth, very ample divisor in a closed Kaehler manifold $X$ then $D$ carries all the homology and homotopy of $X$ below the middle dimension of $X$.
One way to understand this theorem is via Morse theory (the Andreotti-Frankel approa... | 4 | https://mathoverflow.net/users/2356 | 21068 | 13,961 |
https://mathoverflow.net/questions/20699 | 4 | Let $f\_1,\ldots f\_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials of degree below $h(m,l)$ for some $h$ dependent only on $m$ and $l$. Hrushovski gives two references for the truth of... | https://mathoverflow.net/users/5036 | Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields). | In his paper *Constructions in algebra* (MR[349648](http://www.ams.org/mathscinet-getitem?mr=349648)), Seidenberg fixes some errors in Grete Hermann's classic paper *Die Frage der endlich vielen Schritte in der Theorie der Polynomideale* (MR[1512302](http://www.ams.org/mathscinet-getitem?mr=1512302)). Most of the work ... | 1 | https://mathoverflow.net/users/2000 | 21069 | 13,962 |
https://mathoverflow.net/questions/21023 | 7 | Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with residue field $k$ and a flat scheme $\mathcal X$ over $R$, such that
$$ \mathcal X \otimes\_R k \simeq X.$$
In words: Ther... | https://mathoverflow.net/users/5273 | Liftability of Enriques Surfaces (from char. p to zero) | This may not be exactly the answer you are looking for: I and Nick
Shepherd-Barron have an unpublished (so far) proof of liftability in
characteristic $2$, the only non-trivial case. To atone for the fact that I
refer to unpublished results I give a quick sketch of proof.
The proof starts by showing that in a family ... | 8 | https://mathoverflow.net/users/4008 | 21070 | 13,963 |
https://mathoverflow.net/questions/21076 | 7 | Let $R$ be a Dedekind domain with fraction field $K$.
Say that a Dedekind domain $R$ has the **Riemann-Roch property** if: for every nonzero prime ideal $\mathfrak{p}$ of $R$, there exists an element $f \in (\bigcap\_{\mathfrak{q} \neq \mathfrak{p}} R\_{\mathfrak{q}}) \setminus R$, i.e., an element of $K$ which is i... | https://mathoverflow.net/users/1149 | Do all Dedekind domains have the "Riemann-Roch property"? | Yes. Given a maximal ideal $P$ there exists $x \in K \backslash R\_P$. Let $S$ be the finite set of maximal ideals $Q$ so that $x \notin R\_{Q}$. For each $Q \in S$ such that $Q \neq P$ let $y\_Q \in Q\backslash P$. The element $f$ given by multiplying $x$ by large positive powers of all the $y\_Q$ has the desired prop... | 7 | https://mathoverflow.net/users/519 | 21079 | 13,969 |
https://mathoverflow.net/questions/21071 | 14 | This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and do calculations for toy examples. It feels like the answer should be (more) obvious to those who play around with finite... | https://mathoverflow.net/users/763 | Can we bound degrees of complex irreps in terms of the average conjugacy class size? | Yes, take 2-extraspecial group $2^{2n+1}$, plus or minus should not matter. It has $2^{2n}$ irreducible representations of degree 1 and one of degree $2^n$. So your $K\_G$ is about 2 while $d(G)=2^n$.
| 12 | https://mathoverflow.net/users/5301 | 21091 | 13,976 |
https://mathoverflow.net/questions/21086 | 4 | Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be finitely generated? If not, is there a simple counterexample?
| https://mathoverflow.net/users/5229 | When are intersections of finitely generated field extensions finitely generated? | As Brian Conrad remarked above, subextensions of finitely generated extensions are also finitely generated. Here is a prove. I wish there would be a simpler one!
* If $L/K$ is a finitely generated field extension and $L'$ an intermediate field, then $L'/K$ is also finitely generated.
Proof: Since $tr.deg\_K(L) = tr... | 13 | https://mathoverflow.net/users/2841 | 21093 | 13,978 |
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