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https://mathoverflow.net/questions/11699 | 12 | I am posting my comment from [this question](https://mathoverflow.net/questions/8853/what-assumptions-and-methodology-do-metaproofs-of-logic-theorems-use-and-employ) as a separate question, as was recommended to me.
(EDIT: I'm sorry if it ended up being too similar a question, I just wanted to phrase it in the termino... | https://mathoverflow.net/users/1916 | Where are we working when we prove metamathematical theorems? | There are many flavors of "meta" in logic. Most make very minimal use of the metatheory. For example, the Montague Reflection Principle in Set Theory says the following:
>
> **Metatheorem.** For every formula $\phi(x)$ of the language of Set Theory, the following is provable: If $\phi(a)$ is true then there is an ... | 11 | https://mathoverflow.net/users/2000 | 11756 | 7,968 |
https://mathoverflow.net/questions/11761 | 13 | Is there any special reason that we use the sines and cosines functions in the Fourier Series, while we know that if we chose any maximal orthonormal system in L2, we would get the same result?
Is it something historical or what?
Thanks in advance.
| https://mathoverflow.net/users/3124 | Why sin and cos in the Fourier Series? | $1$. Mathematical reason.
There is one reason which makes the basis of complex exponentials look very natural, and the reason is from complex analysis. Let $f(z)$ be a complex analytic function in the complex plane, with period $1$.
Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$... | 14 | https://mathoverflow.net/users/2938 | 11763 | 7,973 |
https://mathoverflow.net/questions/11768 | 7 | Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either analytically or algebraically in terms of Kahler differentials. It can also be defined as the topological genus of $X$ c... | https://mathoverflow.net/users/290 | Reference for equivalent definitions of the genus | For $\mathrm{dim} H^0(X, \Omega^1\_X) = \dim H^1(X, \mathbb{Q})$ see <http://en.wikipedia.org/wiki/Hodge_theory>. For $\dim H^1(X, \mathbb{Q}) =$ number of tori use induction and the Mayer-Vietoris sequence.
(And for $\mathrm{dim} H^0(X, \Omega^1\_X) = \mathrm{dim} H^1(X, \mathcal{O}\_X)$ see <http://en.wikipedia.org... | 7 | https://mathoverflow.net/users/nan | 11769 | 7,976 |
https://mathoverflow.net/questions/11684 | 6 | Take $W = S\_n$ for simplicity, though other Weyl groups work too. Let $r\_i$ denote the $i$th simple reflection acting on ${\mathbb A}^n$, and $\partial\_i = 1/(x\_i - x\_{i+1}) ({\operatorname{Id}} - r\_i)$ denote the corresponding divided difference operator.
It's easy to show that the operators $r\_i + c \partial... | https://mathoverflow.net/users/391 | Reference for representation of Weyl group using r_ + c∂_ | The difference operators (as you presumably know) are defined in a paper of Bernstein–Gelfand–Gelfand on Schubert cells etc. which is probably roughly as old as you can get. The fact that the $r\_i + c\partial\_i$ satisfy the Coxeter relations is implied by (equivalent to, pretty much) the fact that the graded/degenera... | 4 | https://mathoverflow.net/users/1878 | 11770 | 7,977 |
https://mathoverflow.net/questions/11758 | 10 | Is there an algorithm for solving the following problem: let $g\_1,\ldots,g\_n$ be permutations in some (large) symmetric group, and $g$ be a permutation that is known to be in the subgroup generated by $g\_1,\ldots,g\_n$, can we write $g$ explicitly as a product of the $g\_i$'s?
My motivation is that I'm TAing an in... | https://mathoverflow.net/users/622 | Algorithm for decomposing permutations | Yes. The general rule of thumb is that groups described by permutations are computationally easy, groups described by generators and relations have computational problems that are generally undecidable, and matrix groups are somewhere in between.
There's a whole book "Permutation group algorithms" by Seress, Cambridg... | 11 | https://mathoverflow.net/users/440 | 11783 | 7,983 |
https://mathoverflow.net/questions/11781 | 4 | Is it possible to compute complex powers in finite fields? Given a $\in \mathbb{F}\_p$ ($p$ prime), how can one compute $a^i$ per example?
| https://mathoverflow.net/users/3264 | Complex powers in finite fields | I certainly don't think that for arbitrary complex numbers, you'll be able to define a notion of exponentiation (take $\pi$ for example). However, I think if you restrict to algebraic numbers, you'll have more of a chance. Here's something I cooked up for (sometimes) handling an $n$th root of unity $\omega$:
I assume... | 16 | https://mathoverflow.net/users/1916 | 11788 | 7,987 |
https://mathoverflow.net/questions/11798 | 28 | I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks.
I ask that because I will start to learn about neural networks and machine learning on my own to help in the analysis I am doing on my PhD about patterns of genome evolution.
Thank y... | https://mathoverflow.net/users/1776 | Mathematics for machine learning | For basic neural networks (i.e. if you just need to build and train one), I think basic calculus is sufficient, maybe things like gradient descent and more advanced optimization algorithms. For more advanced topics in NNs (convergence analysis, links between NNs and SVMs, etc.), somewhat more advanced calculus may be n... | 11 | https://mathoverflow.net/users/3035 | 11807 | 8,002 |
https://mathoverflow.net/questions/11774 | 35 | For the definitions of the equivalence relations on algebraic cycles see <http://en.wikipedia.org/wiki/Adequate_equivalence_relation>.
I want to know how far away from each other the equivalence relations on algebraic cycles are and what the intuition is for them.
My impression is that rational equivalence gives mu... | https://mathoverflow.net/users/nan | difference between equivalence relations on algebraic cycles | I will focus on complex projective varieties.
Codimension one
---------------
The situation in codimension one is considerably simpler than in higher codimensions.
Codimension one rational equivalence classes are parametrized by $Pic(X)= H^1(X,\mathcal O\_X^{\ast})$ while algebraic equivalence classes are paramet... | 37 | https://mathoverflow.net/users/605 | 11813 | 8,005 |
https://mathoverflow.net/questions/3269 | 50 | Serre's *A Course in Arithmetic* gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (8n + 7)$ : first one shows that the condition is necessary, which is straightforward. To show it is sufficient, a lem... | https://mathoverflow.net/users/290 | Intuition for the last step in Serre's proof of the three-squares theorem | The intuition for this method of passing from a rational solution to an integral solution seems pretty simple to me: passing from a rational solution to a nearby integral point (not necessarily a solution) is passing to a point whose denominators are 1, so you can anticipate that when you intersect the line through you... | 34 | https://mathoverflow.net/users/3272 | 11822 | 8,009 |
https://mathoverflow.net/questions/11821 | 2 | Can anyone give me an explicit isomorphism between $SU(2)$ and the three sphere?
What about for higher spheres? This question [link text](https://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient) seems to indicate that there exists a homeomorphism from $SU(n)/SU(n-1)$ to the $(2n-1)$-sphere.... | https://mathoverflow.net/users/1977 | $SU(2)$ and the three sphere | Elements of $SU(2)$ look like this:
$$ x = \begin{pmatrix} a & - \overline{b} \\ b & \overline{a} \end{pmatrix},$$
where $|a|^2 + |b|^2 = 1$. This follows easily from $x^\* = x^{-1}$. So you map that matrix to the point $(a,b)$ in $\mathbb{C}^2$, and this is your diffeomorphism.
| 8 | https://mathoverflow.net/users/703 | 11823 | 8,010 |
https://mathoverflow.net/questions/11364 | 20 | This question is inspired by [Cohomology of fibrations over the circle](https://mathoverflow.net/questions/4361/cohomology-of-fibrations-over-the-circle) Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points were not addressed there. So I decided ... | https://mathoverflow.net/users/2349 | Cohomology of fibrations over the circle: how to compute the ring structure? | This is a continuation of Ryan's answer above, but it has become too large for a comment. I wanted to work out the details of Ryan's example explicitly, so that we can see explicitly where your conditions fail to determine the cohomology; perhaps this can help you to pin down precisely what conditions you want. It does... | 15 | https://mathoverflow.net/users/250 | 11826 | 8,013 |
https://mathoverflow.net/questions/11828 | 16 | Recently I've been reading J.P. May's [A Concise Course in Algebraic Topology](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf). In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate subgroups H and H' such that H is properly contained in H'... | https://mathoverflow.net/users/303 | two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation. | The subgroup $H=\mathbb Z^\mathbb N$ of $G\_1=\mathbb Z^\mathbb Z$ is mapped to a proper subgroup by translation. By considering the semidirect product $G\_1\rtimes\mathbb Z$, you can make translation on $G\_1$ an inner automorphism.
The first theorem on p.26 tells you that if $G=\pi(B,b)$, $H=p(\pi(E,e))$, $H\_1=p'(... | 10 | https://mathoverflow.net/users/2035 | 11834 | 8,018 |
https://mathoverflow.net/questions/11827 | 5 | Looked around a bit and couldn't seem to find a similar question. (either that or it was worded with vocabulary above the multivariable calculus I've taken. :))
Roughly worded: I would like to develop an algorithm (either in the form of "action to take each discrete time step" or "do these actions at exactly these ti... | https://mathoverflow.net/users/3275 | Navigation solution for frictionless vehicles. | The optimal solution is given by the proportional navigation guidance law, see for example the Wikipedia page: [on proportional navigation](http://en.wikipedia.org/wiki/Proportional_navigation). This solution applies for the general case where the target point B is moving. In our case it is fixed which doesn't change t... | 1 | https://mathoverflow.net/users/1059 | 11847 | 8,028 |
https://mathoverflow.net/questions/11816 | 6 | The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg Kuperberg said that GL(n,C)/U(n) is a CAT(0) space. I was wondering why this is true, or if there's a reference for this.
| https://mathoverflow.net/users/2669 | Why is GL(n,C)/U(n) a CAT(0) space? | It also suffices to check that the sectional curvature of this space, using the Riemannian metric induced by the Killing form, is nonpositive. I recommend that you both figure out how to do the explicit calculation of the sectional curvature for this particular example and learn the general theory referred to in Andy's... | 2 | https://mathoverflow.net/users/613 | 11862 | 8,041 |
https://mathoverflow.net/questions/11621 | 16 | Let $X$ be a smooth projective variety over $F$, and $E/F$ - finite Galois extension.
There is an extension of scalars map $CH^\\*(X) \to CH^\\*(X\_E)$. The image lands in the Galois invariant part of $CH^\\*(X\_E)$, and in the case of rational coefficients, all Galois-invariant cycles are in the image (EDIT: this foll... | https://mathoverflow.net/users/2260 | Obstructions to descend Galois invariant cycles | Let us keep the notations from above, and let's write $G:=\mathrm{Gal}(E/F)$. Let me quickly recall the origin of the Brauer obstruction: it really comes from the Hochschild-Serre spectral sequence
$$H^p(G,E^q(X\_E,\mathbf{G}\_m))\Longrightarrow E^{p+q}(X,\mathbf{G}\_m)$$
(I'm writing $E^{\ast}=H^{\ast}\_{\mathrm{e... | 11 | https://mathoverflow.net/users/3049 | 11863 | 8,042 |
https://mathoverflow.net/questions/11845 | 3 | Automata theory is mainly concerned with Turing machines and all its relatives-in-spirit. $\lambda$-calculus is rather rarely mentioned in textbooks on automata theory.
What's the common name of the theory mainly concerned with $\lambda$-calculus and its relatives? (I think, "mathematical logic", "computability theor... | https://mathoverflow.net/users/2672 | Theory mainly concerned with $\lambda$-calculus? | ["combinatory logic"](http://plato.stanford.edu/entries/logic-combinatory/)
| 4 | https://mathoverflow.net/users/2361 | 11876 | 8,050 |
https://mathoverflow.net/questions/11904 | 8 |
>
> **Question:** Is there a polynomial map from $\Bbb R^n$ to $\Bbb R^n$ under which the image of the positive orthant (the set of points with all coordinates positive) is all of $\Bbb R^n$ ?
>
>
>
Some observations:
My intuition is that the answer must be 'no'... but I confess my intuition for this sort of g... | https://mathoverflow.net/users/2502 | A polynomial map from $\Bbb R^n$ to $\Bbb R^n$ mapping the positive orthant onto $\Bbb R^n$? | The map $z\in\mathbb C\mapsto z^4\in\mathbb C$, when written out in coordinates, is a polynomial map which sends the *closed* first quadrant to the whole of $\mathbb R^2$---and by considering cartesian products you get the same for $\mathbb R^{2n}=\mathbb C^n$.
**Later:** as observed in a comment by [Charles](https:/... | 16 | https://mathoverflow.net/users/1409 | 11905 | 8,071 |
https://mathoverflow.net/questions/11908 | 1 | This question is motivated by my most spectacular [answer](https://mathoverflow.net/questions/10239/is-it-true-that-as-z-modules-the-polynomial-ring-and-the-power-series-ring-over/10240#10240) on MO (:
Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\... | https://mathoverflow.net/users/2083 | Torsion-free and torsionless abelian groups | If by finite you mean finitely presented, then the answer is no. For instance, Let $A = \mathbb{Z}[x]/(2x-1) = \mathbb{Z}[\frac12]$. Then, like $\mathbb{Q}$, $\text{Hom}(A,\mathbb{Z}) = 0$.
| 5 | https://mathoverflow.net/users/1450 | 11910 | 8,073 |
https://mathoverflow.net/questions/11899 | 6 | In [wikipedia](http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29#Sheaves), sheaves were first defined in the case of concrete categories (with usual identity and gluing axioms), then in the general case. (writing it as an "exact" sequence)
Do these two definitions agree? I find the definition for concrete categori... | https://mathoverflow.net/users/nan | Definition of sheaves in wikipedia | I think you're quite right; Wikipedia's "concrete definition" is only correct for concrete categories whose underlying-set functor is (not just faithful but) conservative, i.e. such that any morphism which is a bijection on underlying sets is an isomorphism in the category. The page does say that the concrete definitio... | 10 | https://mathoverflow.net/users/49 | 11911 | 8,074 |
https://mathoverflow.net/questions/11026 | 27 | Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental groups of 3-dim manifolds etc. However, the book is from 1992, so much of the material summarizes research done by Cannon, Th... | https://mathoverflow.net/users/2192 | Automatic groups - recent progress | I'm not an expert in the area, but here's a few highlights:
Bridson [distinguished automatic and combable groups.](https://www.ams.org/mathscinet-getitem?mr=2016694)
Burger and Mozes found examples of [biautomatic simple groups.](https://www.ams.org/mathscinet-getitem?mr=1839489)
Mapping class groups were origina... | 14 | https://mathoverflow.net/users/1345 | 11915 | 8,078 |
https://mathoverflow.net/questions/11907 | 2 | I am fitting a linear regression line to my data and computing the confidence interval for some predicted $y$ (independent variable) (<http://people.stfx.ca/bliengme/ExcelTips/RegressionAnalysisConfidence2.htm>).
Now I want to do the inverse. I need a way to measure the confidence for some predicted $x$ (dependent va... | https://mathoverflow.net/users/3285 | Linear Regression Confidence Interval | Usually you predict a dependent variable y and calculate a confidence interval, i.e. given x0, you calculate [y-, y+] where y will probably lie in.
For the reverse, if you have a y0 and want to find [x-, x+] for whatever reasons, regression will not help.
The appropriate tool for this kind of analysis could be stru... | 1 | https://mathoverflow.net/users/1313 | 11921 | 8,083 |
https://mathoverflow.net/questions/11737 | 0 | I designed a kernel function (to be used within SVM) which has the expression $tr(AB)$ in it. For efficient implementation of this, I was wondering if I could write $tr(AB)$ as an inner product: $\phi(A)^T \phi(B)$? What is the function $\phi()$?
| https://mathoverflow.net/users/3244 | Get rid of tr() in SVM kernel trick | Matus is right. But if the matrices $A$, and $B$ have certain properties like being symmetric, or diagonal, then simply just vectorizing the matrices and taking their inner product would be equal to the $tr(AB)$.
| 0 | https://mathoverflow.net/users/3287 | 11924 | 8,084 |
https://mathoverflow.net/questions/11934 | 15 | I recently stumbled across this number, and then (foolishly, most likely) decided to try to describe it in a blog post
<http://frothygirlz.com/2010/01/14/big-numbers-part-2/>
Q - Are there any comparisons of Graham's Number, hell, even G1, to other well known "big" numbers, such as googolplex?
I'd just like to ha... | https://mathoverflow.net/users/3290 | Magnitude of Graham's Number? | I think all that can really be said is that a googolplex is much, much smaller.
This isn't a direct comparison of a googolplex with Graham's number, but maybe it will help give some perspective:
Some back-of-the-envelope/Mathematica calculations tell me that
$10^{(10^{100})}\approx 3^{(3^{(3^{4.86})})}$
and so... | 17 | https://mathoverflow.net/users/1916 | 11936 | 8,090 |
https://mathoverflow.net/questions/11932 | 18 | I find the following description of Artin motives in [Wikipedia](http://en.wikipedia.org/wiki/Motive_%28algebraic_geometry%29%23Tannakian_formalism_and_motivic_Galois_group). Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in MO to pr... | https://mathoverflow.net/users/2938 | References for Artin motives | André's book is the main reference for the "yoga" of motives. You'll find a description of Artin motives in the Voevodsky formalism in
Beilinson and Vologodsky - <http://www.math.uiuc.edu/K-theory/0832/>
Wildehaus - <http://www.math.uiuc.edu/K-theory/0918/>
From the tannakian view point, Artin motives are just r... | 6 | https://mathoverflow.net/users/1985 | 11938 | 8,092 |
https://mathoverflow.net/questions/11939 | 4 | Given a finite graph $\Gamma$, one has the
[right-angled Artin group](http://en.wikipedia.org/wiki/Right-angled_Artin_group#Right-angled_Artin_groups) $A(\Gamma )$. Its generators $s\_1, \dots s\_n$ bijectively correspond to vertices of $\Gamma$ and the relators are $s\_is\_j=s\_js\_i$ provided the corresponding vert... | https://mathoverflow.net/users/1573 | "Remove a vertex" map for right-angled Artin groups | No. Let $\Gamma$ be the graph with two vertices and no edges - the non-abelian free group of rank two - and let $g$ be the commutator of the two generators $s\_1$ and $s\_2$. Then $g$ is certainly non-trivial, but $g$ dies whenever you kill $s\_1$ or $s\_2$.
UPDATE:
For an example with a connected graph, let's take... | 4 | https://mathoverflow.net/users/1463 | 11940 | 8,093 |
https://mathoverflow.net/questions/11929 | 6 | I was hoping that someone could help clarify a source of confusion for me, I must be doing and saying something wrong but I just don't know what:
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let
$$L(s,E)=\sum\_{n=1}^{\infty}a\_n(E)n^{-s}$$
be the Hasse-Weil $L$-function of $E$. Finally, let $\tilde{E}$ be the r... | https://mathoverflow.net/users/3289 | Alternate expresion of L-series coefficients | There are two different recursions involved here, one for the points of $E$ over ${\mathbb F}\\_{p^n}$, and the other for the coefficients of the $L$-function.
If we write $a\_p = \alpha + \beta,$ where $\alpha\beta = p$ (so $\alpha$ and $\beta$ are
the two roots of the char. poly. of Frobenius), then
$$1 + p^n - ... | 4 | https://mathoverflow.net/users/2874 | 11944 | 8,096 |
https://mathoverflow.net/questions/11916 | 12 | In [Theory mainly concerned with lambda-calculus?](https://mathoverflow.net/questions/11845/theory-mainly-concerned-with-lambda-calculus/11861#11861), F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:
>
> That would never stick unless there's another good reason. Besides, the ... | https://mathoverflow.net/users/3154 | Is functional programming a branch of mathematics? | So, I'm a computer scientist working in this area, and my sense is the following:
You cannot do good work on functional programming if you are ignorant of the logical connection, period. However, while "proofs-as-programs" is a wonderful slogan, which captures a vitally important fact about the large-scale structure... | 10 | https://mathoverflow.net/users/1610 | 11976 | 8,115 |
https://mathoverflow.net/questions/11977 | 2 | Let $G$ be a group and $X$ a set equipped with a *transitive* right $G$-action. Further, let $c: X\to X$ be a $G$-equivariant map. Is it true that $\text{Stab}(x) = \text{Stab}(c(x))$ for all $x\in X$?
This doesn't seem to be an interesting mathoverflow question on its own, but the reason I ask is the following: In H... | https://mathoverflow.net/users/3108 | Equivariant map preserves stabilizer | That's essentially the same as [this question](https://mathoverflow.net/questions/11828). So the answer is negative: if $X=H\backslash G$, the stabilizer of $Ha\in X$ is
$a^{-1}Ha$; if $b$ is any element such that $bHb^{-1}\subset H$, the map
$$Ha\mapsto Hba$$ is $G$-equivariant; if $bHb^{-1}\ne H$, we get a counterex... | 7 | https://mathoverflow.net/users/2653 | 11982 | 8,117 |
https://mathoverflow.net/questions/11974 | 25 | According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relations while in Universal Algebra we only allow functions.
Also, we know that Category Theory generalized Universal Algebra. From wikipedia:
>
> Block... | https://mathoverflow.net/users/2701 | Is there a relationship between model theory and category theory? | Between model theory and category theory broadly conceived: not anything really compelling, because a category, on its own, does not stand as an interpretation for anything.
Between model theory and categorical logic, however: yes, I think the overlap is large.
A spot of history: the man most deserving, in my opin... | 7 | https://mathoverflow.net/users/3154 | 11985 | 8,119 |
https://mathoverflow.net/questions/8930 | 3 | In my quest to understand all things Spearman, consider the following problem:
Given random variable $x$ with known variance, $\sigma^2$, and $p \in [-1,1]$, one can construct a random variable $y$ such that the Pearson correlation coefficient of the variables $x$ and $y$ is exactly $p$. (Let $y = p x + \sqrt{(1-p^2)... | https://mathoverflow.net/users/2570 | construct random variable with a fixed level of Spearman Coefficient to another | You don't even need $\sigma^2$ to construct such a variable.
Let $Z$ be $+1$ with probability $q$, and $-1$ with probability $1-q$, and independent of $X$.
Let $Y = e^XZ$. This squashes X to the positive reals preserving order, and then may change the sign.
$y\_1$ and $y\_2$ have the same ordering as $x\_1$ and $... | 3 | https://mathoverflow.net/users/2954 | 11995 | 8,127 |
https://mathoverflow.net/questions/11752 | 2 | In this post I want to look at an issue I was in doubt when looking at the comment of F. G. Dorais in the post [In model theory, does compactness easily imply completeness?](https://mathoverflow.net/questions/9309/in-model-theory-does-compactness-easily-imply-completeness)
F. G. Dorais remark was:
>
> Blockquote
>... | https://mathoverflow.net/users/2701 | In what sense Fraissean view point shows Model Theory can be done without any formal syntax and deduction rule? | After your answer, I think I understand better where you see a problem. I don't think you fully appreciate the way of interpreting formulas from Fraïssé's point of view. For simplicity, I will follow your lead and stick with the case of a language with just one relation symbol. It's not hard to generalize, but that wou... | 3 | https://mathoverflow.net/users/2000 | 11998 | 8,129 |
https://mathoverflow.net/questions/215 | 19 | In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of Frobenius acting on the $H^i\_\mathrm{et}(X, Q\_\ell)$ are algebraic integers with complex absolute value $q^{i/2}$.
For s... | https://mathoverflow.net/users/81 | Equivalent statements of the Riemann hypothesis in the Weil conjectures | The reason why that inequality is equivalent to RH (for curves) is the *functional equation*.
The polynomial $L(z)$ I speak about below would arise in practice as the numerator of the zeta-function of the curve, with $z = q^{-s}$ and the usual version of the Riemann hypothesis for $L(q^{-s})$ is equivalent to the sta... | 20 | https://mathoverflow.net/users/3272 | 12001 | 8,131 |
https://mathoverflow.net/questions/12003 | 5 | In going from Riemann surface theory to the theory of algebraic curves over fields $k$ that are not necessarily $\mathbb{C}$, I would like to understand more about how the notion of a covering map carries over.
If I have a compact, connected Riemann surface $M$, a cover of $M$ by another such Riemann surface, say $N$... | https://mathoverflow.net/users/3310 | Covering maps of Riemann surfaces vs covering maps of $k$-algebraic curves | No, the property of having small neighbourhoods whose preimage is a disjoint union
of $n$ homeomorphic open sets does not hold in the Zariski topology (once $n > 1$, i.e.
the cover is non-trivial). The reason is
that non-empty Zariski open sets are always very big; in the case of a curve,
their complement is always ju... | 10 | https://mathoverflow.net/users/2874 | 12004 | 8,133 |
https://mathoverflow.net/questions/11948 | 9 | Say a ring $R$ is an isolated hypersurface singularity if $R = k[x\_1, \ldots, x\_n]\_{(x\_1, \ldots, x\_n)}/(W)$, where $k$ is a field and $W \in k[x\_1, \ldots, x\_n]$ is such that the ideal $(\partial\_1 W, \ldots, \partial\_n W)$ is $(x\_1, \ldots, x\_n)$-primary. For finitely generated $R$-modules $M$ and $N$ defi... | https://mathoverflow.net/users/3293 | Isolated hypersurface singularities, Chow groups and D-branes | Assume $k= \mathbf C$ and $W$ homogeneous. Let $X=Proj (k[x\_1,\cdots,x\_n]/(W))$. $X$ is then a smooth hypersuraface in $\mathbb P\_{n-1}$.
Assume $n=2d$ is even. Corollary 3.10 of the paper you quoted says that $\theta=0$ for all pairs iff the homological Chow group $CH^{d-1}\_{hom}$ modulo $[h]^{d-1}$ is not torsi... | 8 | https://mathoverflow.net/users/2083 | 12010 | 8,137 |
https://mathoverflow.net/questions/12045 | 66 | The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
| https://mathoverflow.net/users/3319 | What are fixed points of the Fourier Transform | The following is discussed in a little more detail on pages 337-339 of Frank Jones's book "Lebesgue Integration on Euclidean Space" (and many other places as well).
Normalize the Fourier transform so that it is a unitary operator $T$ on $L^2(\mathbb{R})$. One can then check that $T^4=1$. The eigenvalues are thus $1$,... | 86 | https://mathoverflow.net/users/317 | 12047 | 8,164 |
https://mathoverflow.net/questions/12009 | 70 | One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders $a\_i$ where $a\_i|a\_{i+1}$ and a free abelian group).
The proof that I know, and am not entirely happy with goes as fol... | https://mathoverflow.net/users/66 | Is there a slick proof of the classification of finitely generated abelian groups? | I reject the premise of the question. :-)
It is true, as Terry suggests, that there is a nice dynamical proof of the classification of finite abelian groups. If $A$ is finite, then for every prime $p$ has a stable kernel $A\_p$ and a stable image $A\_p^\perp$ in $A$, by definition the limits of the kernel and image o... | 49 | https://mathoverflow.net/users/1450 | 12053 | 8,169 |
https://mathoverflow.net/questions/11978 | 43 | I was very surprised when I first encountered the [Mertens conjecture](http://en.wikipedia.org/wiki/Mertens_conjecture). Define
$$ M(n) = \sum\_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the Riemann Hypothesis, which is equivalent to $M(n) = O(n^{\frac12 + \epsi... | https://mathoverflow.net/users/2954 | Heuristically false conjectures | I think this example fits, in 1985 H. Maier disproved a very reasonable conjecture on the distribution of prime numbers in short intervals. The probabilistic approach had been thoroughly examined by Harald Cramer. Nice paper by Andrew Granville including this episode in (mathematical) detail, page 23 (or 13 out of 18 i... | 19 | https://mathoverflow.net/users/3324 | 12063 | 8,176 |
https://mathoverflow.net/questions/7074 | 11 | Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be related to R being a rational singularity.
Lipman showed that if X is a desingularization of Spec(R), then one has an exact... | https://mathoverflow.net/users/2083 | Class groups of normal domains over finite fields | As requested in the comments, here's an example of a local, normal $2$-dimensional domain R in positive characteristic such that $\mathrm{Cl}(R)$ is not torsion: choose an elliptic curve $E \subset \mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the local ring at the origin of the affin... | 26 | https://mathoverflow.net/users/986 | 12073 | 8,181 |
https://mathoverflow.net/questions/12065 | 0 | Hi, I want to be able to solve a linear program that has constraints that are either zero or a range. An example below in LP\_Solve-like syntax shows what I want to do. This doesnt work. In general all the decision variables Qx can be 0 or a <= Qx <= b where a > 0 and b > a. All decision variables must be integers.
`... | https://mathoverflow.net/users/3323 | linear program with zeros | You can use the following modelling trick to transform you problem in a integer linear program: for each constraint of the type
$$ Q\_i = 0 \text{ or } L\_i \le Q\_i \le M\_i$$
(on an integer variable $Q\_i$) introduce a new binary variable $B\_i$ and write
$$ L\_i B\_i \le Q\_i \le M\_i B\_i $$
If, as in your exampl... | 1 | https://mathoverflow.net/users/1184 | 12075 | 8,182 |
https://mathoverflow.net/questions/12095 | 4 | Let $k|\mathbb{F}\_q$ be a field extension. An $\mathbb{F}\_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}\_q$-subalgebra $A \_0$ of $A$ such that $A \_0 \otimes \_{\mathbb{F}\_q} k \cong A$ via the *canonical* morphism $a \otimes \lambda \mapsto a \lambda$.
Now, my question is if this notion can be properly glo... | https://mathoverflow.net/users/717 | F_q-structures on schemes | I think that notion you cite from Digne and Michel is not a good one as you will not get a well-defined Frobenius. I suggest replacing $X\_0$ by a pair $(X\_0,p)$, where $p:X\to X\_0$ is a morphism of $\mathbb{F}\_q$ schemes such that $X$ is a product $X\_0\times\_{\mathbb{F}\_q} \mathrm{Spec}(k)$ via the structure mor... | 7 | https://mathoverflow.net/users/2308 | 12101 | 8,193 |
https://mathoverflow.net/questions/12066 | 10 | Is there some condition on a complex algebraic surface that implies it has a smooth canonical divisor? I am searching for the sharpest possible condition, but sufficient criteria would be nice as well.
For example, the question is quite simple for surfaces that can be embedded in $P^3$. Roughly, just notice that in t... | https://mathoverflow.net/users/3314 | When is the canonical divisor of an algebraic surface smooth? | Any smooth projective surface with nonempty $K\_X$ is obtained by blowing up finitely many points on its unique minimal model. From the formula $K\_X=f^\*K\_Y+E$ for the blowup, you see that the exceptional divisors of the blowup are always in the base locus of $|K\_X|$. Thus, the problem is reduced to the minimal mode... | 8 | https://mathoverflow.net/users/1784 | 12103 | 8,194 |
https://mathoverflow.net/questions/12089 | 3 | Is there an upper bound on the genus of a graph that has a book embedding on say k pages, or can the genus be arbitrarily large? If not a general bound is known, what happens for k=3?
| https://mathoverflow.net/users/3327 | Upper bound on the genus of a k-page graph | Every $n$-vertex genus-$g$ graph has at most $3n + 6g - 6$ edges, by Euler's formula. Now consider the three-page graph in which $n-3$ vertices are connected in a path (in one of the pages, it doesn't matter which one), followed by three vertices that are each connected (in separate pages) to everything in the path. It... | 3 | https://mathoverflow.net/users/440 | 12106 | 8,197 |
https://mathoverflow.net/questions/12100 | 29 | In many formulation of Class Field theory, the Weil group is favored as compared to the Absolute Galois group. May I asked why it is so? I know that Weil group can be generalized better to Langlands program but is there a more natural answer?
Also we know that the abelian Weil Group is the isomorphic image of the rec... | https://mathoverflow.net/users/2701 | Why Weil group and not Absolute Galois group? | One reason we prefer the Weil group over the Galois group (at least in the local case) is that the Weil group is locally compact, thus it has "more" representations (over $\bf C$). In fact, all $\bf C$-valued characters of $Gal(\bar{\bf Q\_p} / \bf Q\_p)$ have finite image, where as that of $W\_{\bf Q\_p}$ can very wel... | 15 | https://mathoverflow.net/users/3247 | 12111 | 8,201 |
https://mathoverflow.net/questions/12109 | 25 | In *The Geometry of Schemes* by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}\_{X,p}$ is reduced for closed points $p \in X$. However, this does not seem to work in general, since $X$ may not have enough closed points. What additional hypot... | https://mathoverflow.net/users/3333 | Reduced scheme and closed points | There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27)
But under some very reasonable additional conditions - I think quasi-compactness will be sufficient, if you are happy with using Zorn's lemma - the result holds. Use/prove the existence of a closed point, and the fact that localizing a re... | 13 | https://mathoverflow.net/users/1107 | 12112 | 8,202 |
https://mathoverflow.net/questions/12092 | 11 | It is well known that Beck's theorem for Comonad is equivalent to Grothendieck flat descent theory on scheme.
There are several version of derived noncommutative geometry. I wonder whether someone developed the triangulated version of Beck's theorem. And What does it mean,if exists?
| https://mathoverflow.net/users/3156 | Is there triangulated category version of Barr-Beck's theorem? | There isn't a descent theory for derived categories per se - one can't glue objects in the derived category of a cover together to define an object in the base. (Trying to apply the usual Barr-Beck to the underlying plain category doesn't help.)
But I think the right answer to your question is to use an enriched ver... | 23 | https://mathoverflow.net/users/582 | 12115 | 8,204 |
https://mathoverflow.net/questions/12119 | 26 | **Background**
Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 \rightarrow F(U) \rightarrow G(U) \rightarrow H(U) \rightarrow 0$ is exact for any open set $U$. My solution to thi... | https://mathoverflow.net/users/1106 | Reverse mathematics of (co)homology? | I don't have Hartshorne, so I can't address the specifics of this case. However, there is a very interesting paper by Andreas Blass *Cohomology detects failures of the Axiom of Choice* (TAMS 279, 1983, 257-269), which addresses questions of this type and should at least put you on the right track.
| 19 | https://mathoverflow.net/users/2000 | 12126 | 8,211 |
https://mathoverflow.net/questions/12129 | 5 | Hello, I'm writting something about Malcev categories and monadicity. The fact is that I need to know if Graph is or not complete (have all finite limits). It seems easy but I would like a real answer (not my feelings saying that it is) and I don't find that information anywhere.
Thank you for your answers.
| https://mathoverflow.net/users/3338 | Category of graphs. | First let me just mention that *complete* usually means "has all small limits". If you want to say "has all finite limits", you could use the term *finitely complete* or just *has finite limits*.
You did not explain which particular category of graphs you are talking about, but luckily almost any reasonable choice wi... | 5 | https://mathoverflow.net/users/1176 | 12131 | 8,214 |
https://mathoverflow.net/questions/11868 | 15 | *Formally étale* means that the infinitesimal lifting property is *uniquely* satisfied. If the map is also locally of finite presentation, then it is called *étale*. One of many characterizations (see EGA 4.5.17) of *étale* is flat and unramified. So my question is whether the weaker condition of *formally étale* still... | https://mathoverflow.net/users/917 | Does formally etale imply flat? | It seems that Anton Geraschenkos answer to a previous question [Is there an example of a formally smooth morphism that is not smooth](https://mathoverflow.net/questions/195/is-there-an-example-of-a-formally-smooth-morphism-which-is-not-smooth) does the trick here as well. His example of a formally smooth map that is no... | 14 | https://mathoverflow.net/users/917 | 12139 | 8,219 |
https://mathoverflow.net/questions/12118 | 15 | Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear algebra in characteristic $R$" should be: it should be the study of the category $R\text{-bimod}$ of $(R,R)$-bimodules. For e... | https://mathoverflow.net/users/78 | What is an algebraic group over a noncommutative ring? | It seems that you want some notions on noncommutative group scheme,right?
In fact, A.Rosenberg has introduced **noncommutative group scheme** in his work with Kontsevich
["noncommutative grassmannian and related constructions"](http://www.mpim-bonn.mpg.de/preblob/3621) (2008). Actually, this work gave a systematically ... | 5 | https://mathoverflow.net/users/1851 | 12141 | 8,220 |
https://mathoverflow.net/questions/12140 | 2 | Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{vmatrix} = |A||B|.$$
The above statement is quite easy to prove using linearity etc properties of the determinant. However t... | https://mathoverflow.net/users/2938 | Statement of Lagrange's theorem on determinants(elementary question). | Let $A\_{S,T}$ denote the submatrix of $A$ with rows indexed by the elements of $S$ and columns
by the elements of $T$; let $A'\_{S,T}$ denote the submatrix of $A$ with rows indexed by the
elements not in $S$ and columns by the elements not in $T$. Then we have an expansion
$$
\det(A) = \sum\_T (-1)^{\omega(S,T)} \de... | 5 | https://mathoverflow.net/users/1266 | 12145 | 8,223 |
https://mathoverflow.net/questions/2703 | 22 | Continuing an amazingly interesting chain of answers about [motivic cohomology](https://mathoverflow.net/questions/2146/whats-the-yoga-of-motives), I thought I should learn about the Beilinson conjectures, referred there.
I have found some references, and they seem to present the conjectures from different sides, e.g... | https://mathoverflow.net/users/65 | Beilinson conjectures | Let me talk about Beilinson's conjectures by beginning with $\zeta$-functions of number fields and $K$-theory. Space is limited, but let me see if I can tell a coherent story.
### The Dedekind zeta function and the Dirichlet regulator
Suppose $F$ a number field, with
$$[F:\mathbf{Q}]=n=r\_1+2r\_2,$$
where $r\_1$ is... | 66 | https://mathoverflow.net/users/3049 | 12148 | 8,224 |
https://mathoverflow.net/questions/12136 | 3 | I’ve couldn’t find any information about the free category built up from that Freyd cover. Where can I find more about the Freyd cover of a category (not a topos!)?
**Edit:** The definition has been given in Lambek and Scott's "Higher order categorical logic". I think (according to L. Román) it is initial among all ... | https://mathoverflow.net/users/3338 | Freyd cover of a category. | I don't know anything about it myself, but here are some other phrases you might try looking up.
The Freyd cover of a category is sometimes known as the **Sierpinski cone**, or "**scone**". It's also a special case of Artin gluing. Given a category $\mathcal{T}$ and a functor $F: \mathcal{T} \to \mathbf{Set}$, the *... | 6 | https://mathoverflow.net/users/586 | 12149 | 8,225 |
https://mathoverflow.net/questions/12154 | 7 | For p a constant in (0,1) and n going to infinity such that pn is an integer,
consider the distribution on n bits that selects a random subset of pn bits, sets those to 1, and sets the others to 0.
What is the largest k = k(n,p) so that the induced distribution on any k bits is 1/10 close in total variation distance (a... | https://mathoverflow.net/users/3343 | Local view of setting p*n out of n bits to 1 | You want $\frac15 = \sum\_t |P\_1(count=t) - P\_2(count=t)|$.
where $P\_1$ has a binomial distribution and $P\_2$ is hypergeometric.
The difference between these distributions is shown in [this Mathematica demonstration](http://demonstrations.wolfram.com/BinomialApproximationToAHypergeometricRandomVariable/).
I ... | 3 | https://mathoverflow.net/users/2954 | 12160 | 8,235 |
https://mathoverflow.net/questions/11923 | 14 | The question was edited several times. Most recent version, suggested by Fedja:
>
> Does there exist an open set $U\subset \mathbb R^n$ `(n>1)` that contains balls of arbitrarily large radius and such that no polynomial mapping $p\colon \mathbb R^n\to\mathbb R^n$ takes $U$ onto $\mathbb R^n$? (Take $n=2$ if you pre... | https://mathoverflow.net/users/2912 | Sets that can be mapped onto R^n by a polynomial | It is enough to construct a sequence of pairwise disjoint disks $D\_j$ of infinite total volume so that the image $\bigcup\_j f(D\_j)$ has $0$ density for any polynomial mapping $f:\mathbb R^2\to\mathbb R^2$. Then you can connect them by very thin passageways to get a topological disk and add just a finite area to the ... | 11 | https://mathoverflow.net/users/1131 | 12166 | 8,239 |
https://mathoverflow.net/questions/12144 | 28 | Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since $BH \cong EG/H$, $G/H$ acts on $BH$ and we can write $BG \cong E(G/H) \times\_{G/H} EG/H$. Thus, we have a fiber bundl... | https://mathoverflow.net/users/947 | Classifying Space of a Group Extension | Yes. The principal bundles are the same and your guess that $BA$ is an abelian group is exactly right. A good reference for this story, and of Segal's result that David Roberts quotes, is Segal's paper:
G. Segal. Cohomology of topological groups, Symposia Mathematica IV (1970), 377- 387.
The functors $E$ and $B$ c... | 10 | https://mathoverflow.net/users/184 | 12167 | 8,240 |
https://mathoverflow.net/questions/12056 | 12 | Background
----------
In [a recent question about Fibonacci numbers](https://mathoverflow.net/questions/11885/nontrivial-question-about-fibonacci-numbers), [it was claimed](https://mathoverflow.net/questions/11885/nontrivial-question-about-fibonacci-numbers/11961#11961) that
>
> every integer can be written in th... | https://mathoverflow.net/users/1079 | Representing numbers in a non-integer base with few (but possibly negative) nonzero digits | This is an answer to your "actual question" (2), building on some of the ideas in Douglas Zare's answer.
**Lemma 1:** Suppose that $0 < r < 1$. Let $S=\lbrace \epsilon r^i : \epsilon = \pm 1 \text{ and } i \in \mathbb{Z}\_{\ge 0} \rbrace$. Fix $k \ge 1$. Let $S\_k$ be the set of sums of the form $s\_1+\cdots+s\_k$ su... | 8 | https://mathoverflow.net/users/2757 | 12177 | 8,248 |
https://mathoverflow.net/questions/12097 | 8 | Is there a notion of a cobordism which is compatible with bundle structure?
That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can be made into a bundle whose bundle structure, when restricted to $E$ or $F$, is the bundle structure of $E$ or $F$.
An... | https://mathoverflow.net/users/3329 | Cobordisms of bundles? | I'll assume you're talking about principal G-bundles. These are classified by maps into $BG$, the base of the universal $G$-bundle, so if we have bundles classified by $f:E \to BG$ and $g:F \to BG$, you are looking for a bordism between $f$ and $g$ - whether there exists a $h : W \to BG$ connecting these classifying ma... | 5 | https://mathoverflow.net/users/2368 | 12179 | 8,249 |
https://mathoverflow.net/questions/12180 | 4 | I am glad to see that a general question like [Is there a relationship between model theory and category theory?](https://mathoverflow.net/questions/11974/is-there-a-relationship-between-model-theory-and-category-theory "this") receives quite a lot attention and no down-votes for being too general and unspecific. So I ... | https://mathoverflow.net/users/2672 | Can infinite first-order categories be specified other than as categories of models? | Sure, by direct construction. Rings, preorders, the category of paths of a given graph, etc. But that's not what you wanted to know, is it?
| 3 | https://mathoverflow.net/users/3154 | 12181 | 8,250 |
https://mathoverflow.net/questions/12080 | 3 | I was playing around with the Shannon Switching Game for some planar graphs, trying to get some intuition for the strategy, when I noticed a pattern. Since I only played on planar graphs, I'll restrict the problem to those for now, but we can also ask the problem for non-planar graphs.
Call a graph feasible if $E \ge... | https://mathoverflow.net/users/2363 | About the Shannon Switching Game | Yes it is true, it follows easily from Tutte's disjoint tree theorem for k=2:
<http://lemon.cs.elte.hu/egres/open/Tutte%27s_disjoint_tree_theorem>
In every partition class with n\_i vertices you can have at most 2n\_i-3 edges, plus the edges between the partitions, but this together still gives only 2n-2-|P| edges, w... | 4 | https://mathoverflow.net/users/955 | 12193 | 8,258 |
https://mathoverflow.net/questions/1960 | 15 | The number of Dyck paths in a square is well-known to equal the catalan numbers:
<http://mathworld.wolfram.com/DyckPath.html>
But what if, instead of a square, we ask the same question with a rectangle? If one of its sides is a multiple of the other, then again there is a nice formula for the number of paths below the ... | https://mathoverflow.net/users/955 | Dyck paths on rectangles | Since that Mirko Visontai told me that the answer is ${a+b\choose a}/(a+b)$ if $\gcd(a,b)=1$. The proof is the following (with k=a and l=b):
The number of 0--1 vectors with $k$ 0's and $l$ 1's is ${k+l\choose k}$, so we have to prove that out of these vectors exactly $1/(k+l)$ fraction is an element of $L(k,l)$. The ... | 10 | https://mathoverflow.net/users/955 | 12196 | 8,261 |
https://mathoverflow.net/questions/12190 | 2 | Has anyone ever seen any papers or books including set-theoretic descriptions of formal language theory? Specifically, I'm interested in how one would formalize context-free grammars with sets.
Some of this, I suppose is fairly obvious. For example, strings would use a foundational formalism much like ordered pairs (... | https://mathoverflow.net/users/3357 | Set-theoretic foundations for formal language theory? | One can do this using less technology, too...
Let $\Sigma$ be an alphabet, $N$ a set of non-terminals, and $\Sigma^\\*$ and $(\Sigma\cup N)^\\*$ the full languages on $\Sigma$ and $\Sigma\cup N$, respectively. A context-free grammar is a finite subset $G\subset N\times(\Sigma\cup N)^\\*$. Given one such grammar $G$ t... | 7 | https://mathoverflow.net/users/1409 | 12201 | 8,265 |
https://mathoverflow.net/questions/12085 | 149 | I would like to ask about examples where experimentation by computers has led to major mathematical advances.
=============================================================================================================
A new look
----------
Now as the question is **five years old** and there are certainly **more e... | https://mathoverflow.net/users/1532 | Experimental mathematics leading to major advances | The Prime Number Theorem was conjectured by Gauss from looking (*very hard*, one can presume...) at a table of the primes $\leq10^6$. It is not with too much effort that one can read his *Disquisitiones* as a set of tricks to determine primality with as little work as possible, and one can understand the motivation: he... | 65 | https://mathoverflow.net/users/1409 | 12206 | 8,270 |
https://mathoverflow.net/questions/12200 | 24 | How does one prove that on $S^n$ (with the standard connection) any geodesic between two fixed points is part of a great circle?
For the special case of $S^2$ I tried an naive approach of just writing down the geodesic equations (by writing the Euler-Lagrange equations of the length function) and solving them to gain... | https://mathoverflow.net/users/2678 | Geodesics on spheres are great circles | Although Jose has made the essentially the same point, I just want to elaborate (this is really just a comment, but I always run out of room in the comment box).
What nobody else has mentioned explicitly is that you *should* have trouble solving the Euler-Lagrange equation for the length functional. The Euler-Lagrang... | 29 | https://mathoverflow.net/users/613 | 12217 | 8,277 |
https://mathoverflow.net/questions/12224 | 3 | If $A = (\alpha\_{ij}) \in \mathbb{C}^{nxm}$ we have simple algorithms by which to determine $\mathrm{rank}(A)$. However, is there a polynomial $f \in \mathbb{C}[\alpha\_{ij}]$ where $f \colon \mathbb{C}^{nxm} \to \mathbb{N}$ such that $f(A) = \mathrm{rank}(A)$?
In general, is it possible to determine if a particular... | https://mathoverflow.net/users/3121 | Rank(A) and other algorithms as a polynomial | The preimage of a natural number under a polynomial map is always closed, yet the set of matrices with rank one is not closed. For example, the sequence $(A\_n)\_{n\geq1}$ in $M\\_2(\mathbb C)$ with $A\_n=\left(\begin{smallmatrix}1/n&0\\\0&0\end{smallmatrix}\right)$ has all its items of rank one, yet its limit has rank... | 5 | https://mathoverflow.net/users/1409 | 12225 | 8,283 |
https://mathoverflow.net/questions/12211 | 19 | Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) also seems to be a very basic and powerful tool but it is apparently not as widely used.
I searched in the literature... | https://mathoverflow.net/users/3365 | Applications of infinite Ramsey's Theorem (on N)? | The following fact has been called "Ramsey's Theorem for Analysts" by H. P. Rosenthal.
>
> **Theorem.** Let $(a\_{i,j})\_{i,j=0}^\infty$ be an infinite matrix of real numbers such that $a\_i = {\displaystyle\lim\_{j\to\infty} a\_{i,j}}$ exists for each $i$ and $a = {\displaystyle\lim\_{i\to\infty} a\_i}$ exists too... | 21 | https://mathoverflow.net/users/2000 | 12228 | 8,285 |
https://mathoverflow.net/questions/12213 | 4 | As is well known, the quantum groups $SU\_q(n)$, amongst others, arise from $R$-matrix solutions of the Yang-Baxter Equation. My question is: For any subalgebra of $GL(n)$, does there exist an $R$-matrix Yang-Baxter solution that q-deforms it?
| https://mathoverflow.net/users/1095 | FTR Quantization for any Subalgebra of $GL(n)$? | There are certainly FRT-type constructions for all the classical simple Lie groups, types A,B,C,D (and a modification of type A to encompass $GL\_n$ instead of $SL\_n$). See e.g. Klymik and Schmudgen, Quantum Groups and Their Representations (I think that's the title). Using the theory of roots, etc. it is also possibl... | 3 | https://mathoverflow.net/users/1040 | 12241 | 8,293 |
https://mathoverflow.net/questions/12249 | 2 | Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup\_{n=1}^\infty A\_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): at least one of the $\bar{A}\_n$ contains a ball.
Baire category theorem gives:
Fact1:
(1) $\Rightarrow$ (2)
... | https://mathoverflow.net/users/3370 | Baire category theorem | Maybe if you allow an exceptional set, as in "For every $x$ outside of some meagre set, there is a $\delta>0$ such that $B(x,\delta)$ is contained in one of the $\overline{A}\_n$"
(Indeed, the set of all $x$ where the above fails is a closed subset of $X$ with empty interior.)
| 2 | https://mathoverflow.net/users/2912 | 12253 | 8,300 |
https://mathoverflow.net/questions/12248 | 8 | I'm probably missing something obvious, but I've been wondering what the motivation is for requiring the components $A\_\mu$ in a local trivialization of a gauge connection on a smooth principal $G$-bundle to lie in $\mathfrak{g}$, the Lie algebra of $G$. I can see that this gives a couple of nice properties; for examp... | https://mathoverflow.net/users/3372 | Gauge connections and Lie algebras? | First, I never liked working with principal bundles; vector bundles seem easier and more natural to me. Second, I never like thinking about abstract principal $G$-bundles. I prefer fixing a representation of $G$ and viewing the principal $G$ bundle as a reduced frame bundle associated with a vector bundle.
So let $E$... | 10 | https://mathoverflow.net/users/613 | 12257 | 8,304 |
https://mathoverflow.net/questions/12260 | 8 | I haven't learned that much about primary decomposition, but from I understand about Dedekind domains, we have that all fractional ideals are invertible and all (plain old) ideals factor uniquely into a product of prime ideals, so that Dedekind domains should satisfy this condition. Are these the only such rings, or is... | https://mathoverflow.net/users/1916 | When does the group of invertible ideal quotients = the free abelian group on the prime ideals? | First some responses to comments above:
One can (and should) define fractional ideals for any integral domain $R$. A fractional $R$-ideal is a nonzero $R$-submodule $I$ of the fraction field $K$ such that there exists $x \in K \setminus \{0\}$ such that $xI \subset R$. The product of two fractional ideals is again a ... | 17 | https://mathoverflow.net/users/1149 | 12264 | 8,308 |
https://mathoverflow.net/questions/12266 | 13 | Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$. Bracket is a second order operator, hence subbundle $E$ needs to be $C^2$.
Are there any generalizations for subbun... | https://mathoverflow.net/users/3375 | Frobenius Theorem for subbundle of low regularity? | Let me conisder the case when the distribution of planes is of codimension 1 and explain why in this case it is enough to have $C^1$ smoothness in order to ensure the existence of the folitation.
In the case when the distribution is of codimension 1, you can formulate Frobenius Theorem in terms of 1-forms. Namely yo... | 14 | https://mathoverflow.net/users/943 | 12269 | 8,312 |
https://mathoverflow.net/questions/12236 | 42 | Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the intersection multiplicity of $Y,Z$ at a generic point $x$ of $Y\cap Z$ as:
$$\sum\_{i\geq 0} (-1)^i\text{length}\_{\mathcal O\_{X... | https://mathoverflow.net/users/2083 | Serre intersection formula and derived algebraic geometry? | There are a number of comments to make about Serre's intersection formula and its relation to derived algebraic geometry.
First, we should be a little more cautious about attribution. The idea of using "derived rings" to give an intrinsic version of the Serre intersection formula is not recent. The idea goes back at ... | 44 | https://mathoverflow.net/users/3049 | 12277 | 8,318 |
https://mathoverflow.net/questions/12169 | 60 | A (discrete) group is **amenable** if it admits a finitely additive probability measure (on *all* its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. But the proof I know is surprisingly convoluted. I'd like to know if there's a more direct proof.
The proof I know ... | https://mathoverflow.net/users/586 | Why are abelian groups amenable? | Here is a simpler argument, combining 1--6 into one step.
Let $G$ be a countable abelian group generated by $x\_1,x\_2,\ldots$. Then a Følner sequence is given by taking $S\_n$ to be the pyramid consisting of elements which can be written as
$a\_1x\_2+a\_2x\_2+\cdots+a\_nx\_n$ with $\lvert a\_1\rvert\leq n,\lvert a... | 28 | https://mathoverflow.net/users/250 | 12281 | 8,321 |
https://mathoverflow.net/questions/12279 | 20 | In modern valuation theory, one studies not just absolute values on a field, but also **Krull valuations**. The motivation is easy enough:
If $k$ is a field, a **valuation ring** of $k$ is a subring $R$ such that for every $x \in k^{\times}$, at least one of $x, x^{-1}$ is an element of $R$. (It follows of course tha... | https://mathoverflow.net/users/1149 | What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations? | Since you asked for it, here is a little bit about the role of valuations in the Lang-Nishimura theorem, one version of which is as follows (my version implies yours):
**Theorem (Lang-Nishimura):** Let $X \to \to Y$ be a rational map between $k$-varieties, where $X$ is integral and $Y$ is proper. If $X$ has a smooth ... | 18 | https://mathoverflow.net/users/2757 | 12287 | 8,326 |
https://mathoverflow.net/questions/12284 | 28 | In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of topological types of such manifolds of dimension 3 (Calabi-Yau threefolds) is bounded. I would like to ask what is known in th... | https://mathoverflow.net/users/2349 | Topologically distinct Calabi-Yau threefolds | This is a very good question, and I would really love to know the answer since its current state seems to be quite obscure. Below is just a collection of remarks, surely not the full answer by any means. I would like to argue that for the moment there is no any deep mathematical reason to think that the Euler number of... | 34 | https://mathoverflow.net/users/943 | 12294 | 8,332 |
https://mathoverflow.net/questions/12303 | 17 | Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the Hasse-Weil conjecture for modular curves. Are there similar applications for Shimura varieties?
| https://mathoverflow.net/users/nan | Why is one interested in the mod p reduction of modular curves and Shimura varieties? | The Eichler-Shimura relation doesn't just prove the Hasse-Weil conjecture for modular curves. It e.g. attaches Galois representations to modular forms of weight 2. More delicate arguments (using etale cohomology with non-constant coefficients, machinery that wasn't available to Shimura) attaches Galois representations ... | 32 | https://mathoverflow.net/users/1384 | 12304 | 8,337 |
https://mathoverflow.net/questions/12226 | 4 | Let $\textbf{HoTop}^\*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^\*$, i.e. pointed homotopy equivalence. All constructions like cone or suspensions are pointed/reduced.
A triangle $X\to Y\to Z\to \Sigma X$ is call... | https://mathoverflow.net/users/3108 | Five lemma in HoTop* and arbitrary pointed model categories | This is false for spaces.
Let $X = S^0, Y = S^1$, and $f:X \to Y$ be the trivial map. Then $Z = Cf$ is $S^1 \vee S^1$. Then $[X,Y]$ is trivial, so then the the truth of this statement would imply: If you have a map $g: Z \to Z$ which is the identity on the first circle, and such that the induced map $S^1 \to S^1$ aft... | 5 | https://mathoverflow.net/users/360 | 12308 | 8,340 |
https://mathoverflow.net/questions/12078 | 5 | I couldn't think of a title for this, but here we go:
Fix $p:S\rightarrow T$, a left fibration of simplicial sets, and an edge $f:\Delta^1 \rightarrow T$. Let $t$ be the first vertex of $f$, and $t'$ be the second vertex.
We name the induced map, $q: S^{\Delta^1}\rightarrow S^{\{1 \}}\times\_{T^{\{1 \}}} T^{\Delta... | https://mathoverflow.net/users/1353 | A question about fibrations of simplicial sets and their fibers | I've got to agree with you; it's not the best-written proof in HTT. Let me go through it glacially slowly (for my own sake!) to see if I can write something that will help clarify the role of $X$.
Let me write $Map(U,V)$ instead of $V^U$. I find it easier to parse on the internet.
First, what is our $X$? It's the s... | 5 | https://mathoverflow.net/users/3049 | 12311 | 8,342 |
https://mathoverflow.net/questions/12102 | 8 | For local field, the reciprocity map establishes almost an isomorphism from the multiplicative group to the Abelian Absolute Galois group. (In global case the relationship is almost as nice). It is tempted to think that there can be no such nice accident.
Do we know any explanation which suggest that there "should b... | https://mathoverflow.net/users/2701 | How natural is the reciprocity map? | The reciprocity map is completely natural (in the technical sense of category theory). For example, if $K$ and $L$ are two local fields,
and $\sigma:K \rightarrow L$ is an isomorphism, then $\sigma$ induces an isomorphism of
multiplicative groups $K^{\times} \rightarrow L^{\times}$ and also of abelian absolute Galois ... | 11 | https://mathoverflow.net/users/2874 | 12318 | 8,348 |
https://mathoverflow.net/questions/12298 | 3 | Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max\_{|z|=1} |Q(z)|$.
So, are there similar results for $k\ge 2$? For example, what is the best $f(n,k)$, such that inequality $||\frac{\partial P}{\partia... | https://mathoverflow.net/users/1888 | Bernstein inequality for multivariate polynomial | Tung, S. H. Bernstein's theorem for the polydisc. Proc. Amer. Math. Soc. 85 (1982), no. 1, 73--76. MR0647901 (83h:32017)
(from MR review): Let $P(z)$ be a polynomial of degree $N$ in $z=(z\_1,\cdots,z\_m)$; suppose that $|P(z)|\leq 1$ for $z\in U^m$; then $\|DP(z)\|\leq N$ for $z\in U^m$ where $\|DP(z)\|^2=\sum\_{i=1... | 3 | https://mathoverflow.net/users/2912 | 12319 | 8,349 |
https://mathoverflow.net/questions/12334 | 12 | This is a sequel to my [earlier question](https://mathoverflow.net/questions/10514/teichmuller-theory-introduction) asking for references for Teichmuller theory and moduli spaces of Riemann surfaces.
In this connection, I have read Chapter 11 of the book [Primer of mapping class groups](https://mathoverflow.net/quest... | https://mathoverflow.net/users/2938 | Teichmuller theory and moduli of Riemann surfaces | One of the main "gains" of the Teichmuller theory approach is that you're dealing with a ball. So you're in a situation where you can readily make analytic arguments using fixed-point theory.
Thurston's homotopy-classification of elements in the mapping class group "reducible, (pseudo) anosov, or finite-order" is on... | 12 | https://mathoverflow.net/users/1465 | 12337 | 8,361 |
https://mathoverflow.net/questions/12327 | 1 | Hi everyone,
I am looking for some ideas (or references) in order to get an explicit SDE (if it exists) which would have a stylised property extending in some sense the mean-reversion property of SDE of Ornstein-Uhlenbeck type.
More formally, is it possible to have a $n$-means reverting process defined by an SDE ?
... | https://mathoverflow.net/users/2642 | Extension of some feature of SDE Ornstein-Uhlenbeck type | I believe your notation is redundant. Let $g = f\_1 + ... + f\_n$. Correct me if I'm wrong, but you just want a function $g$ so that $dS(t) = g(s(t),t)dt + \sigma dW\_t$ has the behavior you specify.
It's not clear exactly what properties you want. One possibility is that you can just let $S$ be a Brownian motion in ... | 2 | https://mathoverflow.net/users/2954 | 12339 | 8,363 |
https://mathoverflow.net/questions/12366 | 9 | In the following suppose L/K is a finite Galois extension of number fields, (maybe it works for other cases also, I don't know) By the Chebotorev density theorem when Gal(L/K) is cyclic, there are infinitely many primes in K that stay inert during this extension (cf Janus p136, Algerbaic Number Fields.) When L/K is non... | https://mathoverflow.net/users/1877 | How many primes stay inert in a finite (non-cyclic) extension of number fields? | If $L/K$ is a finite, Galois extension of number fields such that $\text{Gal}(L/K)$ is not cyclic, then no prime of K remains inert L. Indeed, one always has an isomorphism $D\_p/I\_p\cong \text{Gal}(L\_p/K\_p)$ of the Decomposition group modulo the Inertia group with the Galois group of the corresponding residue field... | 25 | https://mathoverflow.net/users/nan | 12369 | 8,379 |
https://mathoverflow.net/questions/12352 | 21 | In Samuel James Patterson's article titled *Gauss Sums* in *The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae*, Patterson says
"Hecke [proved] a beautiful theorem on the different of k, namely that the class of the absolute different in the ideal class group is a square. This theorem - an anal... | https://mathoverflow.net/users/683 | Context for "Coronidis Loco" from Weil's Basic Number Theory | It is not hard to see that if $L/K$ is an extension of number fields, then the
discriminant of $L/K$, which is an ideal of $K$, is a square in the ideal class group of $K$.
Hecke's theorem lifts this fact to the different. (Recall that the discriminant is the norm of the different.)
If you recall that the inverse di... | 23 | https://mathoverflow.net/users/2874 | 12372 | 8,381 |
https://mathoverflow.net/questions/10671 | 5 | Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F\* denote F with a puncture. Then the space H of representations of pi\_1(F\*) on SU(2) is just SU(2)^2g, and the representation spaces of the two handlbodies sit inside H. Call these spaces Q\_1 and Q\_2 -- we will always think... | https://mathoverflow.net/users/492 | Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds) | It might be helpful to look at the book of Akbulut and McCarthy on Casson's Invariant. I think the answer to Question 1 is fairly clearly explained in Proposition 1.1b of of Chapter III.
| 2 | https://mathoverflow.net/users/3405 | 12379 | 8,386 |
https://mathoverflow.net/questions/11747 | 69 | The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say abo... | https://mathoverflow.net/users/2821 | Galoisian sets of prime numbers | I think it is easiest to illustrate the role of the Langlands program (i.e. non-abelian class field theory) in answering this question by giving an example.
E.g. consider the Hilbert class field $K$ of $F := {\mathbb Q}(\sqrt{-23})$; this is a degree 3 abelian extension of $F$, and an $S\_3$ extension of $\mathbb Q$.... | 81 | https://mathoverflow.net/users/2874 | 12382 | 8,388 |
https://mathoverflow.net/questions/12383 | 3 | Motivation: I was reading through Frenkel's article on geometric Langlands program, and the external tensor product of two perverse sheaves occurred in the definition of the geometric Langlands conjectures. There should be a reference somewhere, but the closest I could find is this research note "Exterior Tensor Produc... | https://mathoverflow.net/users/2623 | External tensor product of two (perverse) sheaves | You don't need to construct $F(U)$ explicitly for $U$ that are not products.
Any point $(a,b)$ of $A\times B$ has a basis of neighbourhoods of the form $A\_1 \times B\_1$,
so the presheaf defined on just these open sets is enough data to sheafify and obtain
the corresponding sheaf. Since sheafification preserves stalks... | 9 | https://mathoverflow.net/users/2874 | 12388 | 8,391 |
https://mathoverflow.net/questions/12377 | 11 | Let $G$ be a reductive group over a finite field (i.e. finite groups over lie type). The case I am most interested in is $G=GL\_{n}(\mathbb{F}\_{q})$; other classical groups are also interesting I think.
Deligne-Lusztig theory has a lot to say about the irreducible representations and characters of these groups. For... | https://mathoverflow.net/users/2623 | Decomposing tensor products of irreducible representations of reductive groups over a finite field | Theorem 1.4.1 in [arxiv:0810.2076](http://arxiv.org/abs/0810.2076) answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of ... | 6 | https://mathoverflow.net/users/1583 | 12391 | 8,393 |
https://mathoverflow.net/questions/12342 | 61 | From time to time, when I write proofs, I'll begin with a claim and then prove the contradiction. However, when I look over the proof afterwards, it appears that my proof was essentially a proof of the contrapositive, and the initial claim was not actually important in the proof.
Can all claims proven by reductio ad... | https://mathoverflow.net/users/1353 | Reductio ad absurdum or the contrapositive? | Although the other answers correctly explain the basic logical equivalence of the two proof methods, I believe an important point has been missed:
* *With good reason*, we mathematicians prefer a direct proof of an implication over a proof by contradiction, when such a proof is available. (all else being equal)
Wha... | 135 | https://mathoverflow.net/users/1946 | 12400 | 8,398 |
https://mathoverflow.net/questions/12370 | 8 | I know this sounds dumb, but I can't for the life of me remember how to expand "TC(x)" into a first-order term in the language of set theory (ZFC, not NBG) where epsilon is the only nonlogical symbol.
The obvious definition is an $\omega$-long sentence $x\cup (\bigcup x)\cup (\bigcup\bigcup x)...$, but that isn't in ... | https://mathoverflow.net/users/2361 | first-order definability transitive closure operator | As Mike Shulman and François G. Dorais correctly point out, the official language of set theory has only the binary relation ε, and so there are no *terms* to speak of in that language beyond the variable symbols.
But no set theorist remains inside that primitive language, and neither is it desirable or virtuous to ... | 15 | https://mathoverflow.net/users/1946 | 12405 | 8,402 |
https://mathoverflow.net/questions/12364 | 5 | I'm not quite sure the best way to ask this, so bear with me: Does anyone know of a subset of integers such that, for any odd prime p, the subset only occupies (p-1)/2 equivalence classes mod p (and does so uniformly)?
For example, take the subset of squares. Elementary number theory shows that they (as quadratic res... | https://mathoverflow.net/users/3400 | Integer subset that only occupies (p-1)/2 equivalence classes mod p? | See section 4.3 of Helfgott and Venkatesh, ["How small must ill-distributed sets be?"](http://math.stanford.edu/~akshay/research/hs.pdf)
for an example of a subset of [1..N] of size about log N with small projections onto Z/pZ,
and section 4.2 for a "guess" about what such subsets might look like in general. They sp... | 6 | https://mathoverflow.net/users/431 | 12407 | 8,404 |
https://mathoverflow.net/questions/12394 | 26 | There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.
Suppose $X$ is a set equipped with a metric $d$. I had initially assumed there must be an $n$ such that $X$ embeds isometrically into $\mathbb{R}^n$, but the followi... | https://mathoverflow.net/users/2510 | Representability of finite metric spaces | Since the paper referred to by Hagen Knaf is published by Springer, it may not be available to one and all. The (publicly viewable) MathSciNet reference is: [MR355836](http://www.ams.org/mathscinet-getitem?mr=355836).
It's a very short paper (7 pages) and the main theorem is:
**Theorem** A metric space can be embed... | 23 | https://mathoverflow.net/users/45 | 12409 | 8,405 |
https://mathoverflow.net/questions/12410 | 5 | Many people are familiar with the notion of an additive category. This is a category with the following properties:
(1) It contains a zero object (an object which is both initial and terminal).
This implies that the category is enriched in pointed sets. Thus if a product $X \times Y$ and a coproduct $X \sqcup Y$ ex... | https://mathoverflow.net/users/184 | Terminology: Is there a name for a category with biproducts? | One name that I have seen used is [semiadditive category](http://ncatlab.org/nlab/show/biproduct#semiadditive_categories_3).
| 7 | https://mathoverflow.net/users/49 | 12419 | 8,412 |
https://mathoverflow.net/questions/12416 | 22 | First of all, let me apologize in advance for the terseness of this question.
It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting theorems over totally real fields. For example, it is now known that many elliptic curves over totally real fields are... | https://mathoverflow.net/users/1464 | The difficulties in proving modularity lifting theorems over non-totally real fields | Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.
---
The most basic difficulty is that there is not a map $R \ri... | 22 | https://mathoverflow.net/users/2874 | 12429 | 8,416 |
https://mathoverflow.net/questions/12428 | 8 | In a two-column double complex, one gets from the associated spectral sequence short exact sequences $0\to E\_2^{1,n-1}\to H^n\to E\_2^{0,n}\to 0$, where $H^n$ is the cohomology of the total complex, but I have never seen the construction of this sequence. Any text I've seen merely states it as a fact, or leaves it as ... | https://mathoverflow.net/users/1481 | How does one get the short exact sequence in a two-column spectral sequence? | This follows precisely from the very definition of convergence of the spectral sequence, once one has identified the $\infty$-term. It is done with some details in McLeary's *User Guide*---which is, in my opinion, a very good reference for both the technicalities and the pragmatics of dealing with spectral sequences.
... | 14 | https://mathoverflow.net/users/1409 | 12430 | 8,417 |
https://mathoverflow.net/questions/12426 | 38 | **Background**
Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being the set of natural numbers, so $\in$ will be some binary relation on the natural numbers.
>
> Can such a relati... | https://mathoverflow.net/users/3410 | Is there a computable model of ZFC? | The Tennenbaum phenomenon is amazing, and that is totally correct, but let me give a direct proof using the idea of [computable inseparability](http://en.wikipedia.org/wiki/Effectively_separable).
**Theorem**. There is no computable model of ZFC.
Proof: Suppose to the contrary that M is a computable model of ZFC. T... | 36 | https://mathoverflow.net/users/1946 | 12434 | 8,420 |
https://mathoverflow.net/questions/12436 | 48 | I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
\*\*
Question
\*\*
Are there any well-motivated introductions to scheme theory?
My idea of what "well-motivated" means are specific enough that I think it warrants a detailed exampl... | https://mathoverflow.net/users/1106 | Motivation for concepts in Algebraic Geometry | I would say that the book you're looking for is probably "The Geometry of Schemes" by Eisenbud and Harris. It is very concrete and geometric, and motivates things well (though I don't think it does so in quite the detail of proving that a topological space is Hausdorff iff $X\to 1$ is separated, but I believe it does d... | 33 | https://mathoverflow.net/users/622 | 12437 | 8,422 |
https://mathoverflow.net/questions/12432 | 11 | This was asked as part of an [earlier question](https://mathoverflow.net/questions/10966/orientability-and-orientation-for-a-differentiable-manifold). But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as you define fundam... | https://mathoverflow.net/users/2938 | Meaning of orientation/orientability over rings other than the integers | I just wanted to mention that while orientability for cohomology with arbitrary coefficients is governed solely by cohomology with coefficients in ℤ, there are other cohomology theories for which is is not true. For example, if you have an action of $\pi\_1(X)$ on an abelian group M, then you can talk about (co)homolog... | 10 | https://mathoverflow.net/users/360 | 12441 | 8,424 |
https://mathoverflow.net/questions/12438 | 20 | I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory.
It seems the defintions and methods go back to K.-T. Chen. The integrals seem to converge like an exponential series. He published many pa... | https://mathoverflow.net/users/2938 | Understanding iterated integrals | The theory of iterated integral gives a mixed Hodge structure on rational homotopy of a variety. In the case of the fundamental group, as far as I know, one can only detect the
nilpotent completion of the fundamental group. (At least, this is the only part of $\pi\_1$ that people work with in a motivic context --- see ... | 14 | https://mathoverflow.net/users/2874 | 12445 | 8,426 |
https://mathoverflow.net/questions/12448 | 8 | As an amusement at the start of [this talk](http://videolectures.net/sicgt07_rosenfeld_falkp/), Moshe Rosenfeld poses the following question.
>
> Suppose that there are n salmon which
> begin at distinct points on a unit
> circle, each facing either clockwise
> or counterclockwise. On a signal, each salmon moves... | https://mathoverflow.net/users/2559 | Moshe Rosenfeld's Salmon Problem | Ah, irony. Now that I've publicly asked the question, I practically trip over a reference.
I just found [this paper](http://amc.imfm.si/index.php/amc/article/view/25/43) of Rosenfeld from 2008, which has large overlap with the talk mentioned in my original post. In the paper it is shown that there are initial configu... | 3 | https://mathoverflow.net/users/2559 | 12450 | 8,429 |
https://mathoverflow.net/questions/12443 | 3 | It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book Elliptic Curves, Section 20.4.
My question is, is there an analogous formula for the Eisenstein series over congruence subgroups? This see... | https://mathoverflow.net/users/3416 | Eisenstein series and the Kronecker limit theorem | Yes, there is a generalization. It was done by Larry Joel Goldstein, in the paper "Dedekind sums for a Fuchsian group". The paper has two parts and was published in the Nagoya Journal in around 1973-74.
| 3 | https://mathoverflow.net/users/2938 | 12458 | 8,434 |
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