question_id string | site string | title string | body string | link string | tags list | votes int64 | creation_date timestamp[s] | comments list | comment_count int64 | category string | diamond int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|
101 | mathoverflow | Supercompact and Reinhardt cardinals without choice | A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
_Definition._ A cardinal $\kappa$ is _supercompact_ if for all ordinals $\lambda$, there exists a transitive inner model $M_\lambda$ of $V$ and an elementary embedding $j_\lambda: V\rightarrow... | https://mathoverflow.net/questions/94900/supercompact-and-reinhardt-cardinals-without-choice | [
"lo.logic",
"large-cardinals",
"set-theory",
"axiom-of-choice"
] | 27 | 2012-04-22T18:53:23 | [
"The point here is that we require $j(\\kappa)$ to be the same in all of these embeddings. Clearly if there is a class embedding its restrictions satisfy this property. And now we can quantify over all $\\lambda$ if need be.",
"I know that this is quite old by now, and I'm sure that Noah and several others here w... | 23 | Science | 0 |
102 | mathoverflow | Orders in number fields | Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an unramified prime in $K$. Is it true that the number of orders in $R$ of index equal to $p^r$, for some natural number $r$... | https://mathoverflow.net/questions/130089/orders-in-number-fields | [
"number-fields",
"nt.number-theory",
"algebraic-number-theory"
] | 26 | 2013-05-08T08:50:58 | [
"Thanks everyone. You can actually count the number of subrings of index $p$. Namely, if $r(x) \\mod p$ has $u$ factors of degree $1$ and $w$ factors of degree $2$, and the rest of higher degrees, then the number of subrings of ${\\mathbb Z}/p{\\mathbb Z}[x]/(r(x))$ of index $p$ is ${u \\choose 2} + w$. ",
"The $... | 8 | Science | 0 |
103 | mathoverflow | Where do uncountable models collapse to? | Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be made isomorphic to any countable model $\mathcal{N}\models T$ in the ground model; that is, whether there is some $\mathca... | https://mathoverflow.net/questions/220850/where-do-uncountable-models-collapse-to | [
"lo.logic",
"set-theory",
"model-theory",
"forcing"
] | 26 | 2015-10-13T21:45:52 | [
"I asked Saharon my version of your question, and it turns out to have a relatively easy positive answer. Here is a write-up, which could probably be improved : users.miamioh.edu/larsonpb/extendible.pdf",
"I've been wondering about this too. I don't see that it matters. By the way, Su Gao's \"On automorphism grou... | 9 | Science | 0 |
104 | mathoverflow | Which sets of roots of unity give a polynomial with nonnegative coefficients? | > **The question in brief:** When does a subset $S$ of the complex $n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in $\mathbb R[z]$ with nonnegative coefficients?
Some trivial necessary conditions include $1\not\in S$ and also that $S$ is self-conjugate so that... | https://mathoverflow.net/questions/214784/which-sets-of-roots-of-unity-give-a-polynomial-with-nonnegative-coefficients | [
"nt.number-theory",
"co.combinatorics",
"polynomials",
"additive-combinatorics",
"roots-of-unity"
] | 26 | 2015-08-14T07:23:56 | [
"\"I believe this is the only way to obtain such a polynomial that is a cyclotomic polynomial\": yes, you're correct. The easiest way to see this is plugging in x=1: it's easy to show that $\\Phi_m(1) = \\begin{cases}0 & m=1 \\\\ p & m=p^k \\\\ 1 & \\text{otherwise}\\end{cases}.$ This essentially says there must be... | 5 | Science | 0 |
105 | mathoverflow | Chromatic Spectra and Cobordism | I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with duals generated by a point. There is a geometric realization functor $|-|:\text{Cat}_{(\infty,n)}\rightarrow \text{Gpd}_... | https://mathoverflow.net/questions/184261/chromatic-spectra-and-cobordism | [
"homotopy-theory",
"stable-homotopy",
"infinity-categories",
"cobordism",
"chromatic-homotopy"
] | 26 | 2014-10-12T17:49:07 | [] | 0 | Science | 0 |
106 | mathoverflow | Planar minor graphs | The theorem of Robertson-Seymour about graph minors says that there exists no infinite family of graphs such that none of them is a minor of another one.
Apparently, it came as a generalization of the Kruskal's theorem that states that there exists no infinite family of rooted ordered trees such that none is a minor o... | https://mathoverflow.net/questions/92421/planar-minor-graphs | [
"graph-minors",
"graph-theory"
] | 26 | 2012-03-27T16:28:06 | [
"@PierreDehornoy: I think it would improve the question if \"true for planar graphs\" was replaced with 'true for plane graphs', because what you write immediately afterwards suggests that you are thinking about planar graphs together with a specified embedding into the plane, and that is called a plane graph.",
... | 5 | Science | 0 |
107 | mathoverflow | Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group | Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real motivation for this question, it just came up in conversation and no one knew the answer. There does not appear to be an... | https://mathoverflow.net/questions/262745/is-every-p-group-the-mathbbf-p-points-of-a-unipotent-group | [
"ag.algebraic-geometry",
"gr.group-theory",
"finite-groups",
"algebraic-groups",
"nilpotent-groups"
] | 26 | 2017-02-20T19:26:35 | [
"Do you require $G$ to be connected?",
"@DrorSpeiser the Zariski closure of a finite set is itself (so not unipotent). I don't see any reason why we should expect any given embedding of D_8 in a unipotent group to come from the F_2-points of a connected subgroup.",
"Take $a=1+E_{12}+E_{34}+E_{56}$, $b=1+E_{23}+... | 19 | Science | 0 |
108 | mathoverflow | Does the Tate construction (defined with direct sums) have a derived interpretation? | Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the tensor product $W \otimes_G M$ with a particular (typically unbounded) complex of projective $\Bbb Z[G]$-modules which happ... | https://mathoverflow.net/questions/234577/does-the-tate-construction-defined-with-direct-sums-have-a-derived-interpretat | [
"at.algebraic-topology",
"homological-algebra",
"group-cohomology"
] | 25 | 2016-03-26T13:01:27 | [
"Another characterisation of Tate cohomology $\\hat H^n(G, M)$ is as $Hom(\\mathbf Z, M[n])$, where the Hom-set is in the category of singularities, defined as the quotient of the usual derived category $D^b(Mod_{\\mathbf Z G})$ modulo its subcategory of perfect complexes. A natural extension thus seems to be (I re... | 3 | Science | 0 |
109 | mathoverflow | Are amenable groups topologizable? | I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (<http://arxiv.org/abs/1210.7895>) - a discrete group $G$ is topologizable iff there exists a topology on $G$ which makes it into a Hausdorff non-discrete topological group.
> **Main... | https://mathoverflow.net/questions/114688/are-amenable-groups-topologizable | [
"gr.group-theory",
"gn.general-topology"
] | 25 | 2012-11-27T10:42:34 | [
"I essentially stopped thinking about this, because I don't know how to proceed, but I thought I'd share one idea which at one point I thought was hopeful: G acts on a certain metric space X by isometries - namely fix a mean m on G and define X to be the set of subsets of G up to sets of mean 0. The metric on X is ... | 9 | Science | 0 |
110 | mathoverflow | Caramello's theory: applications | In [this](https://www.laurentlafforgue.org/math/TheorieCaramello.pdf#page=9) text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation):
_In any case, contemporary mathematics provides an example of extraordinarily deep and highly studied equivalence w... | https://mathoverflow.net/questions/332279/caramellos-theory-applications | [
"ag.algebraic-geometry",
"ct.category-theory",
"topos-theory",
"langlands-conjectures",
"applications"
] | 25 | 2019-05-23T03:04:44 | [
"@TimCampion OK, maybe somebody who actually read those papers will comment on them. But who knows, maybe logic will give us the motivic t-structure!",
"The first two publications listed under \"Publications\" on her website are both papers in English written jointly with Lafforgue. In particular, the second appe... | 4 | Science | 0 |
111 | mathoverflow | $\infty$-topos and localic $\infty$-groupoids? | It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting form a topos $T$, constructing a locale $L$ and a surjection $L \rightarrow T$ 'nice enough' (like a proper surjection, o... | https://mathoverflow.net/questions/93517/infty-topos-and-localic-infty-groupoids | [
"ct.category-theory",
"topos-theory",
"infinity-topos-theory",
"locales",
"infinity-categories"
] | 25 | 2012-04-08T13:23:10 | [
"@SimonHenry In classical topos theory, the Diaconescu localic cover theorem gives a connected locally connected geometric morphism. Often people only stress on the fact that it is open.",
"@IvanDiLiberti the stability under pullback of localic morphisms might be useful at some point yes. But the crucial point w... | 8 | Science | 0 |
112 | mathoverflow | Status of the Euler characteristic in characteristic p | In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
> Enfin signalons que la situation en caractéristique positive est loin d'être aussi satisfaisante que dans le cas transcendant. On aimerait pouvoir disposer d'un groupe de Grothendieck `sauvage' des faisceaux constr... | https://mathoverflow.net/questions/135983/status-of-the-euler-characteristic-in-characteristic-p | [
"ag.algebraic-geometry",
"characteristic-p",
"characteristic-classes",
"euler-characteristics"
] | 25 | 2013-07-07T00:32:03 | [
"You've probably seen this already, but I'm adding it here anyway for others: arxiv.org/abs/1510.03018 by Takeshi Saito is relevant."
] | 1 | Science | 0 |
113 | mathoverflow | 0's in 815915283247897734345611269596115894272000000000 | > Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end?
Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way to remember them would be to come up with a mnemonic. He asked me what word he should come up with if the digit is 0.... | https://mathoverflow.net/questions/393993/0s-in-815915283247897734345611269596115894272000000000 | [
"nt.number-theory",
"open-problems",
"digits"
] | 25 | 2021-05-28T09:10:15 | [
"Maybe it will easy to solve in binary first. Where satisfied for 0!, 1!, 2!, 3! and 4!.",
"The point is that the question is not about the zeros in this number (which is a trivial question) but about the zeros in infinitely many other numbers.",
"I prefer to keep an air of mystery about 81591528324789773434561... | 23 | Science | 0 |
114 | mathoverflow | What's the smallest $\lambda$-calculus term not known to have a normal form? | For Turing Machines, the question of halting behavior of small TMs has been well studied in the context of the Busy Beaver function, which maps n to the longest output or running time of any halting n state TM. The smallest TM with unknown halting behavior has 5 states. It takes $5\cdot2\cdot\log_2(5\cdot4 + 4)$ or rou... | https://mathoverflow.net/questions/353514/whats-the-smallest-lambda-calculus-term-not-known-to-have-a-normal-form | [
"lo.logic",
"lambda-calculus"
] | 24 | 2020-02-25T03:18:56 | [
"Yes, I meant unknown in ZFC. I'll make that explicit.",
"Does \"unknown\" mean \"unknown in ZFC\"? I infer that this is the case from the upper bound being a reference to a program which is known to halt under the assumption of a rank-into-rank cardinal.",
"I mean \"unknown halting behaviour\". We know that (\... | 14 | Science | 0 |
115 | mathoverflow | conjectures regarding a new Renyi information quantity | In a recent paper <http://arxiv.org/abs/1403.6102>, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open conjectures, and I would like to pose them as open questions to the mathematics community on this forum. Perhaps there is a si... | https://mathoverflow.net/questions/172684/conjectures-regarding-a-new-renyi-information-quantity | [
"pr.probability",
"linear-algebra",
"it.information-theory",
"matrix-analysis",
"quantum-mechanics"
] | 24 | 2014-06-25T20:21:44 | [
"There has now been even more progress on Conjecture 2 that is useful enough for applications in quantum information theory. The idea was to make use of the Hadamard three-line theorem (in particular, Riesz-Thorin interpolation). This is detailed in the following paper: arxiv.org/abs/1505.04661 . See also arxiv.org... | 4 | Science | 0 |
116 | mathoverflow | Is A276175 integer-only? | The terms of the sequence [A276123](https://oeis.org/A276123), defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}a_{n-1}-a_{n-2}-1$).
But is it also true for the sequence [A276175](https://oeis.org/A2761... | https://mathoverflow.net/questions/248604/is-a276175-integer-only | [
"nt.number-theory",
"sequences-and-series",
"integer-sequences",
"cluster-algebras"
] | 24 | 2016-08-30T02:54:11 | [
"@YCor the sequences are examples of Y-frieze patterns, which are naturally associated to Fock and Goncharov's $\\mathcal{X}$-cluster varieties.",
"More precisely, it seems that $a^{(5)}$ is still integral (checked for $n<35$), and $a^{(k)}$ is non-integral for $k>5$.",
"Initially I thought all of the recurrenc... | 12 | Science | 0 |
117 | mathoverflow | Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice | While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then wondered whether this was unique. When I got back to my computer, I looked on google and came up with [this](http://camoo.frees... | https://mathoverflow.net/questions/51085/subfields-of-mathbbc-isomorphic-to-mathbbr-that-have-baire-property-w | [
"set-theory",
"gn.general-topology",
"axiom-of-choice",
"fields"
] | 24 | 2011-01-03T19:36:46 | [
"Just a comment - The first link should be camoo.freeshell.org/cfield.isor.html instead of camoo.freeshell.org/cfield.isor to make it more readable.",
"@Ricky: Your translation of DC_omega1 to trees is not correct. The above statement is obviously false, for instance since clearly the one element tree has no cha... | 7 | Science | 0 |
118 | mathoverflow | Example of a quasi-Bernoulli measure which is not Gibbs? | Let $X=\\{0,1\\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only.
A measure $\mu$ is **quasi-Bernoulli** if there is a constant $C\ge 1$ such that for any finite sequences $i,j$, $$ C^{-1} \mu[ij] \le \mu[i]\mu[j] \le C\mu[ij]. $$
(Here as usual $ij$ is the juxtaposition of $i$ and $j$ and $[k]$ is the ... | https://mathoverflow.net/questions/53406/example-of-a-quasi-bernoulli-measure-which-is-not-gibbs | [
"ds.dynamical-systems",
"measure-theory"
] | 24 | 2011-01-26T13:19:29 | [
"Dear Vaughn, dear John, could you give us references about both of your arguments? That would be very useful!",
"Let me point out that there is a standard procedure associating to each almost additive sequence of functions a measure on the algebra up to each level (say $n$ if you consider the element $\\varphi_n... | 3 | Science | 0 |
119 | mathoverflow | Is the Poset of Graphs Automorphism-free? | For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$.
> Is it true that $\mathcal {P}_n$ has no nontrivial automorphisms?
**Remarks:**
This follows if one can recognize a graph... | https://mathoverflow.net/questions/153208/is-the-poset-of-graphs-automorphism-free | [
"co.combinatorics",
"graph-theory",
"posets",
"graph-reconstruction"
] | 24 | 2013-12-30T20:23:03 | [
"Probably has something to do with: math.stackexchange.com/questions/3052384/…",
"Of course there is in fact no need to check, as it anyhow follows from the set edge reconstruction conjecture but I could not find anything about for what values that has been checked.",
"$\\mathcal P_5$ can pretty easily be check... | 4 | Science | 0 |
120 | mathoverflow | An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof? | A famous result in linear algebra is the following.
> An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). However, I want to find an elementary proof avoiding any concepts related to the SNF. My question is: is there an elementary... | https://mathoverflow.net/questions/122345/an-n-times-n-matrix-a-is-similar-to-its-transpose-a-top-elementary-pr | [
"linear-algebra",
"matrices",
"matrix-theory",
"similarity"
] | 24 | 2013-02-19T11:40:49 | [
"After many years this question was first asked, I cannot find anything meeting all of my requirement. Actually, finding Smith Normal Form can be done through elementary row/column operations. So, it is already elementary enough, and I do not have any reason left for avoiding such a simple and powerful tool. Also, ... | 17 | Science | 0 |
121 | mathoverflow | What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra? | There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are invertible superbimodules, and $2$-morphisms are invertible superbimodule homomorphisms to the symmetric monoidal ($\infty$-)gro... | https://mathoverflow.net/questions/186148/what-is-the-symmetric-monoidal-functor-from-clifford-algebras-to-invertible-k-mo | [
"at.algebraic-topology",
"kt.k-theory-and-homology",
"clifford-algebras"
] | 23 | 2014-11-03T18:03:26 | [
"Update: awhile back I asked Lurie this question and his sense is that the functor I want isn't symmetric monoidal but at best lax / oplax.",
"Have you worked out the complex version (graded complex central simple algebras and invertible $KU$-module spectra) of this story? If so, perhaps you can go down to $KO$-m... | 2 | Science | 0 |
122 | mathoverflow | Base change for $\sqrt{2}.$ | This is a direct follow-up to [Conjecture on irrational algebraic numbers](https://mathoverflow.net/questions/173414/conjecture-on-irrational-algebraic-numbers).
Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is there an easy (or, failing that, ... | https://mathoverflow.net/questions/173442/base-change-for-sqrt2 | [
"nt.number-theory",
"ds.dynamical-systems"
] | 23 | 2014-07-06T08:36:56 | [
"@NikitaSidorov I, for one, am completely agnostic on the subject, and have absolutely no clue why anyone would believe what you say everyone believes. I am, of course, just a caveman.",
"@NikitaSidorov And of course, we all speak for ourselves.",
"Of course, we all believe it is transcendental. And of course, ... | 9 | Science | 0 |
123 | mathoverflow | Do all possible trees arise as orbit trees of some permutation groups? | ## I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a compatible left-invariant metric, that is, a metric d inducing the group's topology, and such that d(zx,zy)=d(x,y) for a... | https://mathoverflow.net/questions/61361/do-all-possible-trees-arise-as-orbit-trees-of-some-permutation-groups | [
"gr.group-theory",
"co.combinatorics",
"descriptive-set-theory",
"topological-groups",
"permutations"
] | 23 | 2011-04-11T21:02:55 | [
"a rooted tree is a subset(well, my fault) in \\omega^{<\\omega} that is closed under taking initial segment. For example, the set {(0),(0,0),(0,1),(0,2)} is the tree that has one root and three \"branches\", each terminates at level 1. But we usually think of it intuitively(as drawn on paper). As we fix more and... | 4 | Science | 0 |
124 | mathoverflow | Boundaries of noncompact contractible manifolds | It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of the contractible manifold, and noting that its boundary sphere has the same homology as $B$ by Poincare duality. The "if... | https://mathoverflow.net/questions/89604/boundaries-of-noncompact-contractible-manifolds | [
"gt.geometric-topology",
"mg.metric-geometry"
] | 23 | 2012-02-26T12:43:35 | [
"@User See theorem 11.3 in Massey \"Homology and cohomology theory\", or the section on Poincare duality in Bredon's \"Sheaf theory\". There two references involve (co)homology of general spaces and may be hard for a beginner. Alternatively, solve exercise 35 in section 33 in Hatcher's \"Algebraic topology\".",
... | 17 | Science | 0 |
125 | mathoverflow | Riemannian manifolds etc. as locally ringed spaces? | There are, among others, three general ways of equipping a "space" (which for the purposes of this question could be a topological space or a differentiable manifold, according to the case) with further structure:
(1) "Specifying regular funcions", which leads to locally ringed spaces, e.g. real-analytic manifolds or ... | https://mathoverflow.net/questions/56833/riemannian-manifolds-etc-as-locally-ringed-spaces | [
"locally-ringed-spaces"
] | 23 | 2011-02-27T08:08:01 | [
"From scheme theoretic perspective, how do wish to 'deal' with additional structures like metrics, symplectic forms, hermitian or kahler structures?",
"I'm quite interested in this question. Made any progress since Feb?",
"@JohanesEbert: recovering $g$ up to conformal equivalence would already be nice. And what... | 8 | Science | 0 |
126 | mathoverflow | When does a representation admit a spin structure? | Let $G$ be a finite group, and let $V$ be an $n$-dimensional _real_ representation of $G$. Think of $V$ as given by a homomorphism $$ \rho_V\colon G\to O(n).$$ Write $\chi_V$ for the character of $V$.
Here are two problems.
1. Using _only_ the character $\chi_V$ of $V$, determine whether $\rho_V(G)\subset SO(n)$.
... | https://mathoverflow.net/questions/52643/when-does-a-representation-admit-a-spin-structure | [
"rt.representation-theory"
] | 23 | 2011-01-20T08:10:12 | [
"For 1, it depends what you mean by \"using the character ONLY\". You can't tell the number of elements of each order from the character alone. If you do know the order of each element, then 1 requires just that each element of $G$ has the eigenvalue $-1$ with even (possibly zero) multiplicity, and this is an easy ... | 10 | Science | 0 |
127 | mathoverflow | Is the set of integers of the form $a/(b+c)+b/(a+c)+c/(a+b)$ computable? | The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are [absurdly high](https://web.archive.org/web/20170813162605/https://plus.google.com/+johncbaez999/posts/Pr8LgYYxvbM), namely $$(15447680210874616644195... | https://mathoverflow.net/questions/278747/is-the-set-of-integers-of-the-form-a-bcb-acc-ab-computable | [
"nt.number-theory",
"computability-theory",
"diophantine-equations",
"computer-science"
] | 22 | 2017-08-14T12:16:31 | [
"The goo.gl link in a previous comment points to Quora.",
"mathoverflow.net/questions/264754/…",
"Also discussed at mathematica.stackexchange.com/questions/184956/…",
"Cross-posted to cstheory.stackexchange.com/questions/39383/…",
"(continued) Note that there are odd $n$ for which the associated elliptic cu... | 11 | Science | 0 |
128 | mathoverflow | A mysterious paper of Stallings that was supposed to appear in the Annals | In Stallings's paper
* Stallings, John, _[Groups with infinite products](https://doi.org/10.1090/S0002-9904-1962-10817-9)_ , Bull. Amer. Math. Soc. **68** (1962), 388–389.
he briefly discusses how to prove "several generalizations" of Brown's theorem saying that monotone union of open $n$-cells is an open $n$-cel... | https://mathoverflow.net/questions/405617/a-mysterious-paper-of-stallings-that-was-supposed-to-appear-in-the-annals | [
"reference-request",
"gt.geometric-topology",
"soft-question"
] | 22 | 2021-10-06T13:41:53 | [
"I feel the pain. There's more than one of these in category theory from the 70s-80s, announced to appear in JPAA, but which never turned up.",
"With regards to the publication of this paper, it looks like the journal may have been... stalling",
"@CarloBeenakker: Thanks for noticing that reference! I think tha... | 4 | Science | 0 |
129 | mathoverflow | The multiplication game on the free group | Fix $W\subseteq\mathbb F_2$ and consider the following two-person game: Player 1 and Player 2 **simultaneously** choose $x$ and $y$ in $\mathbb F_2$. The first player wins, say one dollar, iff $xy\in W$. Does this game admit Nash equilibria?
Of course, for very particular choices of $W$ it does, for instance if $W$ i... | https://mathoverflow.net/questions/98493/the-multiplication-game-on-the-free-group | [
"gr.group-theory",
"game-theory",
"amenability"
] | 22 | 2012-05-31T07:19:39 | [
"Ah, that was my misunderstanding.",
"I've edited, saying explicitly that the choices of $x$ and $y$ are simultaneous.",
"Why should P2 win? Notice that the choices are done independently (simultaneously, if you want), so that P2 cannot choose a suitable $y$ such that $xy\\in W$, because (s)he does not know $x$... | 4 | Science | 0 |
130 | mathoverflow | Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$ | $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\tr}{\operatorname{... | https://mathoverflow.net/questions/272355/characterising-critical-points-of-ef-int-m-bigwedge2-df2-textvol | [
"ap.analysis-of-pdes",
"riemannian-geometry",
"calculus-of-variations",
"critical-point-theory"
] | 22 | 2017-06-16T06:50:03 | [
"For the local problem, it may be useful to assume that $M,N$ are both flat in which case you can get rid of the geometry and concentrate on the multilinear algebra."
] | 1 | Science | 0 |
131 | mathoverflow | Smooth thickenings of non-smoothable manifolds | It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold.
**Question 1.**_What can be said about the**smallest** dimension of a smooth manifold that is homotopy equivalent to a given closed topological manifold?_
The following somewhat heavy-handed argument yields a smooth ma... | https://mathoverflow.net/questions/97598/smooth-thickenings-of-non-smoothable-manifolds | [
"gt.geometric-topology"
] | 22 | 2012-05-21T12:43:39 | [
"@Misha: certainly, not all $(n+1)$-dimensional thickenings of closed $n$-manifolds are proper homotopic to $I$-bundles. One can start with an $I$-bundle, and then attach a one-sided $h$-cobordism along the boundary; there are general methods to construct those in higher dimensions. One can also take boundary conne... | 7 | Science | 0 |
132 | mathoverflow | On certain representations of algebraic numbers in terms of trigonometric functions | Let's say that a real number has a _simple trigonometric representation_ , if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or negative) integer powers of $\sin(\cdot)$ at rational multiples of $\pi$ (the number of terms in the product is assumed to b... | https://mathoverflow.net/questions/194181/on-certain-representations-of-algebraic-numbers-in-terms-of-trigonometric-functi | [
"nt.number-theory",
"algorithms",
"algebraic-number-theory",
"computability-theory",
"galois-theory"
] | 22 | 2015-01-17T20:50:44 | [
"You mean solve all minimal polynomials with $\\sin(\\pi\\times rational)$?",
"This is related I guess: mathworld.wolfram.com/TrigonometryAngles.html"
] | 2 | Science | 0 |
133 | mathoverflow | Infinitely many planets on a line, with Newtonian gravity | (I previously asked essentially this [on physics.stackexchange](https://physics.stackexchange.com/questions/56843/infinitely-many-planets-on-a-line-with-newtonian-gravity), but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each integer... | https://mathoverflow.net/questions/128796/infinitely-many-planets-on-a-line-with-newtonian-gravity | [
"mp.mathematical-physics",
"differential-equations"
] | 22 | 2013-04-25T20:31:29 | [
"@Anthony Quas and Jon: I think this is not what he means. He does not \"live in 1D world\" but in 3-space. Just the planets happen to be on the line. The gravity law must be inverse squares. Otherwise the questions make no sense.",
"@AnthonyQuas is correct. The Poisson equation in one dimension in this case is $... | 9 | Science | 0 |
134 | mathoverflow | Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces? | Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \hookrightarrow X$ is a [cofibration](http://ncatlab.org/nlab/show/Hurewicz+cofibration), or that $(X,A)$ is a neighborhoo... | https://mathoverflow.net/questions/134443/are-there-chain-complexes-and-homology-groups-taking-values-in-pairs-of-topo | [
"at.algebraic-topology",
"chain-complexes",
"reference-request",
"gn.general-topology"
] | 22 | 2013-06-21T17:21:30 | [
"@ViditNanda This is two years old by now. Did you make any progress?",
"The pair should be $(X,A)$ of course. ",
"@Vidit: I've sent an email on this. One point is that a pair $(X,S)$ defines filtrations $E_n(X,A)$ which are a base point in dim $0$, $A$ in levels $1$ to $n-1$ and $X$ thereafter. So if $X$ alrea... | 9 | Science | 0 |
135 | mathoverflow | Fake CM elliptic curves | Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy $$ a_p=0, \; \mbox{ for all } \; p \equiv -1 \mod{4}, \; \; \; k/3 < p < k. $$ Can one prove unconditionally that $E$ has CM? This follows from work of ... | https://mathoverflow.net/questions/49937/fake-cm-elliptic-curves | [
"nt.number-theory",
"elliptic-curves"
] | 22 | 2010-12-20T00:09:05 | [
"Dear BCrd : I've pondered that, but I really don't know. My admittedly uneducated guess would be \"probably not\" with the level of precision I'm after....",
"Dear Mike: Can one make \"effective\" (in the spirit of your question) any information coming from the knowledge that Sato-Tate holds in the non-CM case? ... | 2 | Science | 0 |
136 | mathoverflow | Is the equivariant cohomology an equivariant cohomology? | Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology).
$\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A reference is K. Brown's "Cohomology of groups". Namely, $H^\ast_G(X;M)=H^\ast(Hom_{\Bbb Z G}(C_\ast,M))$, where $C_\ast$ i... | https://mathoverflow.net/questions/68330/is-the-equivariant-cohomology-an-equivariant-cohomology | [
"at.algebraic-topology",
"equivariant-homotopy",
"equivariant",
"group-cohomology"
] | 22 | 2011-06-20T17:42:25 | [
"When the action is free, $H^n_G(X;M)\\simeq [X,K(M,n)]_G$; the relative case looks a bit more complicated, $H^n_G(X,Y;M)\\simeq [(X,Y),(K(M,n)\\times BG,BG)]_G$ (see VI.3.11 in the Goerss-Jardine book and projecteuclid.org/euclid.ijm/1256052280). On the other hand, for every $G$-spectrum $E$, $h^{V-W}_G(X):=[S^W\\... | 2 | Science | 0 |
137 | mathoverflow | bound on the genus of a fiber of the Albanese map of a surface with $h^1({\mathcal O})=1$? | This is maybe more an open problem than a question, since I have seriously thought about it and asked several people working on algebraic surfaces with no success. I hope somebody here can suggest an approach different from the standard arguments in surface theory.
BACKGROUND: let $X$ be a smooth minimal complex proje... | https://mathoverflow.net/questions/49169/bound-on-the-genus-of-a-fiber-of-the-albanese-map-of-a-surface-with-h1-mathc | [
"ag.algebraic-geometry",
"algebraic-surfaces"
] | 22 | 2010-12-12T12:12:39 | [
"If $K_S-F$ is effective, than of course I have a bound since $K_S(K_S-F)\\ge 0$ and $K_SF=2g(F)-2$. And by the same argument, I would have a bound if I could determine explicitly an $m$ such that $mK_S-F$ is effective, but that's precisely what I don't know how to do.",
"I GUESS in some sense you can make it. ... | 2 | Science | 0 |
138 | mathoverflow | Row of the character table of symmetric group with most negative entries | The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, the row of the character table of $S_n$ corresponding to the sign representation has the most negative entries of any ro... | https://mathoverflow.net/questions/420865/row-of-the-character-table-of-symmetric-group-with-most-negative-entries | [
"co.combinatorics",
"gr.group-theory",
"rt.representation-theory",
"symmetric-groups",
"characters"
] | 21 | 2022-04-22T09:54:56 | [
"@PerAlexandersson: I don't see any obvious connection to that conjecture.",
"Is this related perhaps to the conjecture that about half of the partition numbers, p(n) are even, as n goes to infinity?",
"@DenisSerre: the sum of the row corresponding to the sign representation is oeis.org/A000700. This quantity g... | 7 | Science | 0 |
139 | mathoverflow | Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity? | _(Copied[from MSE](https://math.stackexchange.com/questions/4472850/can-a-4d-spacecraft-with-just-a-single-rigid-thruster-achieve-any-rotational-v). Offering four bounties over time, I got no response, other than twenty-nine upvotes.)_
It seems preposterous at first glance. I just want to be sure. Even in 3D the behav... | https://mathoverflow.net/questions/448773/can-a-4d-spacecraft-with-just-a-single-rigid-thruster-achieve-any-rotational-v | [
"ca.classical-analysis-and-odes",
"ds.dynamical-systems",
"mp.mathematical-physics",
"differential-equations"
] | 21 | 2023-06-13T06:58:24 | [
"Re, I had not meant to make any adverse changes, and didn't even realise that it made a difference in the rendering. My apologies. I have changed back to back-to-back $$ $$ environments.",
"@LSpice - Now the displayed equations are too closely spaced."
] | 2 | Science | 0 |
140 | mathoverflow | Reference request: deforming a G-local system to a variation of Hodge structure | Let $X$ be a smooth connected quasiprojective variety over $\mathbb{C}$ and let $G$ be a complex reductive group. Let $$\iota: G\to GL_N$$ be a representation and let $$\rho: \pi_1(X(\mathbb{C}))\to G(\mathbb{C})$$ be a homomorphism. I am looking for a reference for the following fact:
> $\rho$ is deformation-equivale... | https://mathoverflow.net/questions/413301/reference-request-deforming-a-g-local-system-to-a-variation-of-hodge-structure | [
"ag.algebraic-geometry",
"complex-geometry",
"hodge-theory"
] | 21 | 2022-01-06T11:59:59 | [
"@SashaP: actually I think this should probably be false over infinite finitely generated fields; I have an idea for a counterexample conditional on Fontaine-Mazur, but it’s maybe too involved for an MO comment.",
"@SashaP: This is a nice question! I think I see an argument for the analogous statement over finite... | 7 | Science | 0 |
141 | mathoverflow | A multiple integral | Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}.$$
Using an indirect method (Landau-Zener formula from the theory of molecular scattering), we had sho... | https://mathoverflow.net/questions/172278/a-multiple-integral | [
"ca.classical-analysis-and-odes",
"real-analysis",
"integration"
] | 21 | 2014-06-20T04:15:04 | [] | 0 | Science | 0 |
142 | mathoverflow | Bounding failures of the integral Hodge and Tate conjectures | It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what this means and whether this can be made more precise.
> > Let $\pi: X \to B$ be a smooth projective morphism of smooth va... | https://mathoverflow.net/questions/168997/bounding-failures-of-the-integral-hodge-and-tate-conjectures | [
"ag.algebraic-geometry",
"nt.number-theory",
"arithmetic-geometry",
"hodge-theory"
] | 21 | 2014-06-04T01:36:23 | [
"It seems unlikely that any of the known counterexamples to the integral Hodge conjecture would give a negative answer to your question. On the other hand, there seems to be no reason to expect a positive answer since the dimension of the space of Hodge classes could jump over infinitely many subvarieties of $B$. I... | 1 | Science | 0 |
143 | mathoverflow | Is the mapping class group of $\Bbb{CP}^n$ known? | In his paper ["Concordance spaces, higher simple homotopy theory, and applications"](https://pi.math.cornell.edu/~hatcher/Papers/ConcordanceSpaces.pdf), Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an understanding of the surgery structure set of the $n$-t... | https://mathoverflow.net/questions/349319/is-the-mapping-class-group-of-bbbcpn-known | [
"reference-request",
"at.algebraic-topology",
"gt.geometric-topology",
"differential-topology",
"mapping-class-groups"
] | 21 | 2019-12-29T07:40:43 | [
"Regarding your last comment: Wall (MR0156354, MR0177421) and Kreck (MR0561244) have computed various mapping class groups of highly connected manifolds up to extension problems.",
"Cool, I wasn't aware of that paper!",
"@skupers Kreck points out to me that the oriented smooth mapping class group $\\pi_0 \\text... | 6 | Science | 0 |
144 | mathoverflow | Cauchy matrices with elementary symmetric polynomials | $\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the [elementary symmetric polynomial](http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial), defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*} e_k(\vx) := \sum_{1 \le l_1 < l_2 < \cdots < l_k \le n} x_{l_1}x_{l_2}\cdots x... | https://mathoverflow.net/questions/114101/cauchy-matrices-with-elementary-symmetric-polynomials | [
"matrices",
"rt.representation-theory",
"linear-algebra",
"co.combinatorics"
] | 21 | 2012-11-21T13:01:30 | [
"Unfortunately, this question has seems to have a negative answer (as proved to me very recently by one of the mathematicians with whom I discussed it; will update this question once I get a chance.)"
] | 1 | Science | 0 |
145 | mathoverflow | p-groups as rational points of unipotent groups | Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, the answer is yes via Witt vectors, but is it true in general?
| https://mathoverflow.net/questions/69397/p-groups-as-rational-points-of-unipotent-groups | [
"p-groups",
"algebraic-groups"
] | 21 | 2011-07-03T06:46:19 | [
"Sorry, after a bit more thought, it occurs to me that, since $p$-groups have centres, it's OK to have the Abelian group as sub-object. Then Theorem 1.8(c) of the paper by Kumar and Neeb shows that $\\operatorname{Ext}^1_{\\text{alg}}(G, \\mathbb G_a) \\cong H^2(\\mathfrak g, \\mathfrak{gl}_1)^{\\mathfrak g}$; but... | 3 | Science | 0 |
146 | mathoverflow | Deciding whether a given power series is modular or not | The degree 3 modular equation for the Jacobi modular invariant $$ \lambda(q)=\biggl(\frac{\sum_{n\in\mathbb Z}q^{(n+1/2)^2}}{\sum_{n\in\mathbb Z}q^{n^2}}\biggr)^4 $$ is given by $$ (\alpha^2+\beta^2+6\alpha\beta)^2-16\alpha\beta\bigl(4(1+\alpha\beta)-3(\alpha+\beta)\bigr)^2=0, $$ where $\alpha=\lambda(q)$ and $\beta=\l... | https://mathoverflow.net/questions/50804/deciding-whether-a-given-power-series-is-modular-or-not | [
"nt.number-theory",
"modular-forms"
] | 21 | 2010-12-31T05:11:02 | [
"@Kevin, thanks for this hint. Because I have no guess about the index of the underlying group in $\\Gamma(1)$, I am not sure that $O(q^{1000})$ would be enough. I didn't try to expand so far (the coefficients grow extremely fast), but what I did (up to $O(q^{50})$) was verifying a possible algebraic relation betwe... | 2 | Science | 0 |
147 | mathoverflow | Schemification (schematization?) of locally ringed spaces | **Motivation:**
Say $F: D \to Sch$ is a diagram in the category of schemes, and we're interested in whether it has a [colimit](http://en.wikipedia.org/wiki/Limit_\(category_theory\)#Colimits_2) (gluings, pushouts, and "categorical" quotients are all examples of colimits). Its colimit $Q$ in the category of locally rin... | https://mathoverflow.net/questions/60524/schemification-schematization-of-locally-ringed-spaces | [
"ag.algebraic-geometry",
"ct.category-theory"
] | 21 | 2011-04-03T23:40:12 | [
"@Andrew: I'm thinking of something like a \"generalized GAGA\", but I'm not really sure what conditions you need on $(X,A)$. I think something like what I'm talking about is covered in chapter 8 of that book, but I really don't know the specifics.",
"@Buschi, Chapter IV seems to be about forming locally ringed ... | 5 | Science | 0 |
148 | mathoverflow | Closed connected additive subgroups of the Hilbert space | It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is the subgroup $L^2(I,\mathbb{Z})$ of the real Hilbert space of all $L^2$ real valued functions on the unit interval $I:=... | https://mathoverflow.net/questions/45322/closed-connected-additive-subgroups-of-the-hilbert-space | [
"hilbert-spaces",
"topological-groups",
"fa.functional-analysis"
] | 21 | 2010-11-08T08:48:41 | [] | 0 | Science | 0 |
149 | mathoverflow | Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$ | A while ago, I came across the following problem, which I was not able to resolve one way or the other.
> Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,x)$ are Lipschitz continuous with respect to $x$ and increasing in $t$. Is it necessarily true that the partial derivati... | https://mathoverflow.net/questions/77957/almost-everywhere-differentiability-for-a-class-of-functions-on-mathbbr2 | [
"ca.classical-analysis-and-odes",
"real-analysis"
] | 21 | 2011-10-12T14:15:06 | [] | 0 | Science | 0 |
150 | mathoverflow | Is the Dieudonne module actually a cohomology group? | One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory
$$M:\left\\{p\text{- divisible groups}/k\right\\}\to \left\\{F\text{-crystals }/k\text{ with slopes in }[0,1]\right\\}.$$
This is supported by the f... | https://mathoverflow.net/questions/235186/is-the-dieudonne-module-actually-a-cohomology-group | [
"arithmetic-geometry",
"crystalline-cohomology"
] | 21 | 2016-04-03T05:24:29 | [
"Doesn't Mazur-Messing interpret the Dieudonne module of a p-divisible group G as rigidified extensions? This suggests that defining a category of G-invariant D-modules should be possible. I would naively guess that the resulting cohomology would be \"generated in degree 1\", just as for abelian varieties, though."... | 2 | Science | 0 |
151 | mathoverflow | Density of first-order definable sets in a directed union of finite groups | This is a generalization of the following [question](https://mathoverflow.net/questions/39798/in-an-inductive-family-of-groups-does-the-probability-that-a-particular-word-is) by John Wiltshire-Gordon.
Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \... | https://mathoverflow.net/questions/43900/density-of-first-order-definable-sets-in-a-directed-union-of-finite-groups | [
"gr.group-theory",
"lo.logic",
"combinatorial-group-theory",
"pr.probability",
"finite-groups"
] | 21 | 2010-10-27T16:10:17 | [
"I agree, cool question!",
"This is a great question.",
"It may also not be true that you will get a density, i.e. that there will be a limit law. I definitely don't know, but Vardi might. Gerhard \"Ask Me About System Design\" Paseman, 2010.11.02",
"@Gerhard: It's not true that you'll always get density z... | 5 | Science | 0 |
152 | mathoverflow | If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well? | Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-dimensional CW complex? If so, how might we construct an example?
A first consideration could be where $X$ has infinitely... | https://mathoverflow.net/questions/267669/if-x-times-y-is-homotopy-equivalent-to-a-finite-dimensional-cw-complex-are-x | [
"at.algebraic-topology",
"homotopy-theory",
"cohomology",
"classifying-spaces"
] | 21 | 2017-04-19T14:52:52 | [
"There are examples by R. Bing and many other mathematicians when a product of a non-manifold B and a real line gives a euclidean space, see e.g. the following papers and references there: ams.org/journals/bull/1958-64-03/S0002-9904-1958-10160-3/… ams.org/journals/proc/1961-012-01/S0002-9939-1961-0123303-2/… Howe... | 6 | Science | 0 |
153 | mathoverflow | Are the eigenvalues of the Laplacian of a generic Kähler metric simple? | It is a [theorem of Uhlenbeck](http://www.jstor.org/stable/2374041) that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set of such metrics is the complement of a [meagre set](http://en.wikipedia.org/wiki/... | https://mathoverflow.net/questions/32810/are-the-eigenvalues-of-the-laplacian-of-a-generic-k%c3%a4hler-metric-simple | [
"dg.differential-geometry",
"complex-geometry",
"riemannian-geometry",
"fa.functional-analysis"
] | 21 | 2010-07-21T08:52:36 | [] | 0 | Science | 0 |
154 | mathoverflow | Homotopy flat DG-modules | A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups $F\otimes_A M$ is acyclic. Homotopy projective and injective DG-modules are defined in the similar way, e.g., a left DG-... | https://mathoverflow.net/questions/40036/homotopy-flat-dg-modules | [
"ct.category-theory",
"homological-algebra",
"differential-graded-algebras",
"flatness"
] | 21 | 2010-09-26T09:53:57 | [
"I think the paper arxiv.org/abs/1607.02609 answers your question.",
"The following paper solves a closely related problem: L.W. Christensen and H. Holm, \"The direct limit closure of perfect complexes\", Journ. of Pure and Appl. Algebra 219, #3, p.449-463, 2015. arxiv.org/abs/1301.0731"
] | 2 | Science | 0 |
155 | mathoverflow | Is there a "direct" proof of the Galois symmetry on centre of group algebra? | Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear automorphism of $\mathbb{Z}[G]^G$, the algebra of $G$-invariant functions on $G$ under convolution, and surprisingly, this induce... | https://mathoverflow.net/questions/415174/is-there-a-direct-proof-of-the-galois-symmetry-on-centre-of-group-algebra | [
"gr.group-theory",
"rt.representation-theory",
"finite-groups",
"characters"
] | 21 | 2022-02-01T13:38:45 | [
"By a generators and relations argument I would think some symbolic argument that holds in any torsion group of exponent coprime to $n$, building $a,b$ from $g,h$ using these assumptions and the assumed $n$th root function. If one could give a group of this type where the result fails, it would rule out such an arg... | 12 | Science | 0 |
156 | mathoverflow | monoidal (∞,1)-categories from weakly monoidal model categories | In _Higher Algebra_ section 4.1.7, Jacob Lurie proves that the underlying $(\infty,1)$-category of a monoidal model category is a monoidal $(\infty,1)$-category.
Dominic Verity and Yuki Maehara have (independently) defined the lax Gray tensor product for model categories $S$ and $T$ for two different models of $(\inft... | https://mathoverflow.net/questions/370172/monoidal-%e2%88%9e-1-categories-from-weakly-monoidal-model-categories | [
"model-categories",
"monoidal-categories",
"infinity-categories"
] | 20 | 2020-08-26T09:02:52 | [
"There was further discussion in chat. To sum up: the suggestion from my second comment was misguided because Lurie really only deals in reflective localizations, and so can't say much about the localization from a model category to its associated $\\infty$-category. But Rune points out that Hinich has results on h... | 9 | Science | 0 |
157 | mathoverflow | Is every positive integer the rank of an elliptic curve over some number field? | For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$?
It's easy to show that the set of such $n$ is unbounded. But can one show that _every_ positive integer is the rank of some elliptic curve over a number field?
The analogous question for a fixed numb... | https://mathoverflow.net/questions/334853/is-every-positive-integer-the-rank-of-an-elliptic-curve-over-some-number-field | [
"nt.number-theory",
"arithmetic-geometry",
"elliptic-curves"
] | 20 | 2019-06-26T07:04:40 | [
"The \"trick\" here is we can use $q \\to\\infty$ results because the question lets us pass to any finite extension. A similar result would follow over number fields from heuristics for the distribution of $L$-functions. In particular, it requires our heuristics to be true only in the 100% setting, and not in the p... | 16 | Science | 0 |
158 | mathoverflow | Non-rigid ultrapowers in $\mathsf{ZFC}$? | _Originally[asked and bountied at MSE](https://math.stackexchange.com/questions/4293034/non-rigid-ultrapowers-in-mathsfzfc):_
> **Question** : Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\omega$ such that the ultra... | https://mathoverflow.net/questions/408178/non-rigid-ultrapowers-in-mathsfzfc | [
"set-theory",
"lo.logic",
"model-theory",
"ultrafilters",
"ultrapowers"
] | 20 | 2021-11-10T08:01:04 | [
"@PaulLarson \" An ultrapower of a structure of infinite Scott height by an ultrafilter on omega shouldn't have its points separated by its Scott process. Should that make it nonrigid?\" I'm not certain, I'm not too familiar with Scott processes.",
"The proof of Harrington's theorem on the Scott ranks of countere... | 7 | Science | 0 |
159 | mathoverflow | A spin extension of a Coxeter group? | Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$.
Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) \in S^2$ satisfying $s \neq t$ and $m_{s,t} < \infty$.
For every $\left( s,t\right) \in \mathfrak{M}$, let $c_{s,t}$ be a... | https://mathoverflow.net/questions/285263/a-spin-extension-of-a-coxeter-group | [
"algebraic-combinatorics",
"coxeter-groups",
"hecke-algebras"
] | 20 | 2017-11-04T12:46:36 | [
"@DavidESpeyer: Thanks -- I've answered the Question positively in the meantime (a year ago), but only in a month or so will probably be able to write up my proof. Meanwhile, I did look into Howlett's and others' Schur-multiplier papers, but never found myself able to get something out of it that wasn't obviously w... | 11 | Science | 0 |
160 | mathoverflow | Isoperimetric inequality and geometric measure theory | The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality:
> **Theorem.** _If $K\subset\mathbb{R}^n$ is compact, then $$ |K|^{\frac{n-1}{n}}\leq n^{-1}\omega_n^{-1/n}\mu_+(K), $$ where $\omega_n$ is the volume of the unit ball and
> $$ \mu_+(K)=\liminf_{h\to 0} ... | https://mathoverflow.net/questions/294882/isoperimetric-inequality-and-geometric-measure-theory | [
"geometric-measure-theory",
"isoperimetric-problems",
"hausdorff-measure"
] | 20 | 2018-03-10T13:32:46 | [] | 0 | Science | 0 |
161 | mathoverflow | Hahn-Banach and the "Axiom of Probabilistic Choice" | Stipulate that the Axiom of Probabilistic Choice (APC) says that for every collection $\\{ A_i : i \in I \\}$ of non-empty sets, there is a function on $I$ that assigns to $i$ a finitely-additive probability measure $\mu_i$ on $A_i$.
Then Hahn-Banach (HB) implies APC, since HB is equivalent to the existence of a finit... | https://mathoverflow.net/questions/281142/hahn-banach-and-the-axiom-of-probabilistic-choice | [
"set-theory",
"axiom-of-choice",
"hahn-banach-theorem"
] | 20 | 2017-09-14T07:57:09 | [
"1. D. Pincus, The strength of Hahn–Banach's Theorem, in: Victoria Symposium on Non-standard Analysis, Lecture notes in Math. 369, Springer 1974, pp. 203-248. Implies Hahn-Banach theorem is weaker than the ultrafilter lemma",
"@Fedor: You might have remembered that HB+SKM (Strong Krein-Milman) imply AC.",
"@Woj... | 8 | Science | 0 |
162 | mathoverflow | Looking for an effective irrationality measure of $\pi$ | Most standard summaries of the literature on [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving any indication of how large qualifies as "sufficiently large." I... | https://mathoverflow.net/questions/210509/looking-for-an-effective-irrationality-measure-of-pi | [
"nt.number-theory",
"diophantine-approximation"
] | 20 | 2015-06-07T11:02:16 | [
"Salikhov's paper is available here: mathnet.ru/links/93c99ab6587bdc1bf0912ec96b1d9f4f/rm9175.pdf",
"I wasn't aware this was migrated, I'll delete my answer in that case.",
"The exponent -42 got by Mahler was quite extraordinary in his time (\"Striking inequality\", Alan Baker). It is valid for all rational p/q... | 5 | Science | 0 |
163 | mathoverflow | Is the determinant of cohomology a TQFT? | If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$ induced by a cobordism from $M$ to $N$?
In some dimensions, there is a natural identification $D(M) = \mathbf{C}$ and y... | https://mathoverflow.net/questions/241807/is-the-determinant-of-cohomology-a-tqft | [
"at.algebraic-topology",
"dg.differential-geometry",
"topological-quantum-field-theory"
] | 20 | 2016-06-09T10:31:23 | [
"I would have thought that the $d$-dimensional theory (or maybe I mean $d\\pm 1$) would assign $\\mathrm{pt} \\mapsto \\mathbb C$-as-an-$E_d$-algebra. But that defines the trivial (framed) theory. Perhaps there's a nontrivial choice of how to descend from a framed theory to an oriented one, but I'm not seeing it.... | 3 | Science | 0 |
164 | mathoverflow | Homeomorphisms of the sphere mapping a geodesic triangulation to another one | Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map $T$ to a geodesic triangulation. What is the homotopy type of $L(S,T)$? A similar question has been solved by [E. Bloch, R. Connelly, D. Henderson, Topology 23 (1984), ... | https://mathoverflow.net/questions/262048/homeomorphisms-of-the-sphere-mapping-a-geodesic-triangulation-to-another-one | [
"gt.geometric-topology",
"riemannian-geometry",
"triangulations"
] | 20 | 2017-02-12T11:27:31 | [
"I agree with the change of title. It is better, but I thought of (S,T) as a structure, not an object; it was ambiguous.",
"I have improved the title",
"The question in the edited title doesn't seem to be the same as the one in the body (and is much easier)."
] | 3 | Science | 0 |
165 | mathoverflow | Polynomials with roots in convex position | Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of $P^{(k)}=\left(\frac{d}{dx}\right)^kP$ are in strictly convex position (ie are the $n-k$ vertices of a convex polygon w... | https://mathoverflow.net/questions/26799/polynomials-with-roots-in-convex-position | [
"gt.geometric-topology",
"cv.complex-variables",
"polynomials"
] | 20 | 2010-06-02T01:21:03 | [
"I have sketched a proof in order to adress the the comment of Denis Serre.",
"I am not sure of why all these real polynomials are in different CCs. Do you pretend that if a coefficient of degree $\\le n-2$ is zero, then $P$ is not in ${\\mathcal P)_n$ ? That seems wrong for the constant coefficient.",
"Ī̲ have... | 3 | Science | 0 |
166 | mathoverflow | On random Dirichlet distributions | Fix a dimension $d\ge2$.
* Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
* For every $\mathbf{x}$ in $Q_d$, let $|\mathbf{x}|=x_1+\ldots+x_d$.
* Let $\Delta_d$ denote the set of points $\mat... | https://mathoverflow.net/questions/55147/on-random-dirichlet-distributions | [
"pr.probability",
"probability-distributions"
] | 20 | 2011-02-11T09:23:01 | [] | 0 | Science | 0 |
167 | mathoverflow | Etale fundamental group of a curve in characteristic $p$ | Let $C$ be a connected, smooth, proper curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\pi_1(C)$ be the etale fundamental group of $C$ - I only care about this as an abstract profinite group, so I omit base points.
It is well known by Grothendieck that $\pi_1(C)$ is topologicall... | https://mathoverflow.net/questions/186687/etale-fundamental-group-of-a-curve-in-characteristic-p | [
"ag.algebraic-geometry",
"fundamental-group"
] | 20 | 2014-11-09T23:09:03 | [
"I think $\\pi_1$ is not known for any hyperbolic curve in any reasonable sense. For instance, just knowing for each prime-to-$p$ cover the $p$-rank of its Jacobian seems very hard - it is not at all obvious that there is a usable finite description of this data.",
"Take $G_1$ an infinitely generated free $p$-gro... | 15 | Science | 0 |
168 | mathoverflow | Finiteness of etale cohomology for arithmetic schemes | By an _arithmetic scheme_ I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
> Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite for all $n \in \mathbb{N}$?
Remarks: Let $j:U \to X$ be the open subset given by removing the fibres of $X \to \mathrm{... | https://mathoverflow.net/questions/284354/finiteness-of-etale-cohomology-for-arithmetic-schemes | [
"ag.algebraic-geometry",
"nt.number-theory",
"arithmetic-geometry",
"etale-cohomology"
] | 20 | 2017-10-25T04:51:27 | [
"@MartinBright: Thanks for the references! I have had a look at the paper of Sato, but it is written in the language of derived categories which are not my forte.... Could you please be more specific which result is most relevant? He seems to have many different purity results in his paper.",
"A couple of referen... | 12 | Science | 0 |
169 | mathoverflow | Infinitely generated non-free group with all proper subgroups free | Is there any example of group $G$ satisfying the following properties?
1. $G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group.
2. $H< G$ implies that $H$ is a free group.
Clearly such an example should be torsion-free and countable.
* * *
(Added, from comments) Fo... | https://mathoverflow.net/questions/351295/infinitely-generated-non-free-group-with-all-proper-subgroups-free | [
"gr.group-theory",
"examples",
"free-groups"
] | 20 | 2020-01-27T14:45:56 | [
"Since a partly equivalent question was just asked, I have added the short argument to discard the uncountable case.",
"@YCor You're probably right I should not have have said \"clearly\" but in my mind it was a very direct application of some known facts (to me at least) in infinite group theory. Swan Theorem sa... | 30 | Science | 0 |
170 | mathoverflow | Characteristic subgroups and direct powers | Solved question: Suppose _H_ is a characteristic subgroup of a group _G_. Is it then necessary that, for every natural number _n_ , in the group $G^n$ (the external direct product of $G$ with itself $n$ times), the subgroup $H^n$ (embedded the obvious way) is characteristic?
Answer: No. A counterexample can be constru... | https://mathoverflow.net/questions/35701/characteristic-subgroups-and-direct-powers | [
"gr.group-theory"
] | 20 | 2010-08-15T18:10:41 | [
"Do you have a reference for \"if $H$ is fully invariant in $G$ then $H \\times H$ is fully invariant in $G \\times G$? Thanks in advance.",
"In the case of finite abelian groups, is it true that $\\operatorname{Aut}(G \\times G \\times G)$ is generated by automorphisms stabilizing one copy of $G$ ? This holds at... | 6 | Science | 0 |
171 | mathoverflow | Large values of characters of the symmetric group | For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the representation and, second, the bound $|\chi(g) |\ \leq \sqrt{ |Z(g)|}$ by the centralizer arising from the formula $\sum_{\chi... | https://mathoverflow.net/questions/441235/large-values-of-characters-of-the-symmetric-group | [
"co.combinatorics",
"rt.representation-theory",
"symmetric-groups",
"characters"
] | 19 | 2023-02-20T06:22:56 | [
"But now suppose we consider a permutation with $n/{3k}$ $k-1$-cycles, $k$-cycles, and $k+1$-cycles, or something like that. The square root of the centralizer is now $\\sqrt{ (n/3k)!^3 (k^3-k)^{n/3k} }$ which is smaller by a factor of roughly $\\sqrt{ 3^{n/k} e^{ - \\pi \\sqrt{2n/3}}}$. So the upper bound you desi... | 8 | Science | 0 |
172 | mathoverflow | What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$? | Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as follows:
$$\mathcal{U}\mathbin{\widehat \star}\mathcal{W}:=\\{A:\\{k: \\{a: a\star k\in A\\}\in\mathcal{U}\\}\in\mathcal{W}\\... | https://mathoverflow.net/questions/427135/what-algebraic-properties-are-preserved-by-mathbbn-leadsto-beta-mathbbn | [
"set-theory",
"lo.logic",
"gn.general-topology",
"model-theory",
"universal-algebra"
] | 19 | 2022-07-22T17:24:11 | [
"@BenjaminSteinberg xy=zw is also preserved. :P",
"One thing that is preserved of course is either xy=x or xy=y",
"Yes. I think chapter 6 talks about free subsemigroups. Of course it doesn't looks at nonassociative magmas. My impression is that few general semigroup identities are preserved.",
"@BenjaminS... | 6 | Science | 0 |
173 | mathoverflow | Mumford-Tate conjecture for mixed Tate motives | Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
1. A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-smooth case via a thickening, as in e.g. Hartshorne's algebraic de Rham cohomology paper),
2. For each embedding $\i... | https://mathoverflow.net/questions/379972/mumford-tate-conjecture-for-mixed-tate-motives | [
"ag.algebraic-geometry",
"nt.number-theory",
"galois-representations",
"mixed-hodge-structure"
] | 19 | 2020-12-29T09:18:09 | [
"@DonuArapura It seems like I might have to, since I have an application in mind!",
"Hi Daniel. I'm not sure it is written down anywhere, so it would be nice if you did."
] | 2 | Science | 0 |
174 | mathoverflow | Is there a classification of reflection groups over division rings? | I asked a [version](https://math.stackexchange.com/questions/3491953/reflection-groups-of-division-rings) of this question in Math StackExchange about a week ago but I've received no feedback so far, so following the advice I received on [meta](https://meta.mathoverflow.net/questions/4416/would-this-question-be-appropr... | https://mathoverflow.net/questions/349809/is-there-a-classification-of-reflection-groups-over-division-rings | [
"gr.group-theory",
"rt.representation-theory",
"division-rings",
"reflection-groups"
] | 19 | 2020-01-06T00:40:10 | [] | 0 | Science | 0 |
175 | mathoverflow | xkcd's "Unsolved Math Problems", straight lines in random walk patterns | STEM student's favourite source of amusement posted a comic titled "Unsolved Math Problems" one of which looks like something that could actually be tackled.
> If I walk randomly on a grid, never visiting any square twice, placing a marble every N steps, on average how many marbles will be in the longest line after NK... | https://mathoverflow.net/questions/406469/xkcds-unsolved-math-problems-straight-lines-in-random-walk-patterns | [
"random-walks"
] | 19 | 2021-10-18T01:49:48 | [
"@Keba if you get boxed in discard that walk and begin again. A better way to think of the question might be: consider ALL possible SAWs of NK steps with a marble placed at every Nth step. What is the average number of marbles on the longest line?",
"@ZachTeitler, re, a (literally) bold new notation for the $5$-a... | 14 | Science | 0 |
176 | mathoverflow | Is there a simpler proof of the key lemma in the paper by Hiroshi Iriyeh and Masataka Shibata on the 3D Mahler conjecture? | In this [remarkable paper](https://arxiv.org/abs/1706.01749) 30 pages are occupied by the proof of the following innocently looking lemma:
Let $K$ be an origin-symmetric convex body in $\mathbb R^3$. There exist three planes through the origin splitting $K$ into $8$ parts of equal volume and such that each two of thes... | https://mathoverflow.net/questions/271972/is-there-a-simpler-proof-of-the-key-lemma-in-the-paper-by-hiroshi-iriyeh-and-mas | [
"gt.geometric-topology",
"convex-geometry"
] | 19 | 2017-06-11T18:05:22 | [
"it might be of interest: in the first part of the proof of Theorem 1 keithmball.files.wordpress.com/2014/11/… Ball constructs a similar map to simplex as Makeev does but he uses Browers Fixed Point Theorem (BFPT) unlike Makeev. So it quite might be that one needs to apply BFPT in a proper way which I don't see yet... | 7 | Science | 0 |
177 | mathoverflow | Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space | Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and in finding the best $c$ such that this is true in a given range, such as $3\leq x\leq X$, say; all of these problems are... | https://mathoverflow.net/questions/237308/checking-mertens-and-the-like-in-less-than-linear-time-or-less-than-sqrtx-s | [
"nt.number-theory",
"analytic-number-theory",
"computational-complexity",
"computation"
] | 19 | 2016-04-25T12:22:32 | [
"Well, numbers of size about $X$ without prime factors between $X^(1/3)$ and $X^{1/2}$ are very roughly as common as the primes - but the sum of the reciprocals of the primes diverges. Or rather - the sum $\\sum_n 1/n$ over all $X\\leq n<2X$ without such prime factors is about a constant times $1/\\log X$, i.e., mu... | 7 | Science | 0 |
178 | mathoverflow | Which manifolds decompose into pants? | In this [nice paper](http://arxiv.org/abs/math/0205011) Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a smooth manifold into some _$(n-1)$-dimensional pair-of-pants_.
Following Mikhalki... | https://mathoverflow.net/questions/113875/which-manifolds-decompose-into-pants | [
"gt.geometric-topology",
"tropical-geometry",
"4-manifolds"
] | 19 | 2012-11-19T13:53:05 | [
"I think that the following holds: let $X$ be obtained by gluing $n$ Pairs-of-Pants, and suppose that $X'$ is obtained by the same gluing rules, except that you add a Dehn twist in one of the gluing faces. Then, in analogy with the Lickorish-Wallace theorem, $X'$ should be obtained from $X$ by a single surgery alon... | 4 | Science | 0 |
179 | mathoverflow | Can a number be palindromic in more than 3 consecutive number bases? | _$2017:$ Was initially asked on [MSE](https://math.stackexchange.com/questions/2234587/can-a-number-be-palindrome-in-4-consecutive-number-bases)_ \- but wasn't solved or updated there since.
**Update $2019$:** I've returned to this problem, made some progress and updated the post here.
(I've basically rewritten this... | https://mathoverflow.net/questions/268590/can-a-number-be-palindromic-in-more-than-3-consecutive-number-bases | [
"nt.number-theory",
"diophantine-equations",
"palindromes"
] | 19 | 2017-04-29T08:32:32 | [
"@GerryMyerson Regarding a relevant sequence, I think A279093 is updated with all known 3-palindromic solutions.",
"Loosely related: oeis.org/A214425",
"@joro Update: Now it is known that the $3$ digit solution does not exist. I will have to update this post.",
"@JoseBrox If we have $n$ and $n+3$, and we also... | 12 | Science | 0 |
180 | mathoverflow | A Linear Order from AP Calculus | In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\\{x^\alpha (\ln{x})^\beta\\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $g$ weakly dominates $f$, and write $f\preceq g$, if
$$\lim_{x\to\infty}\frac{f(x)}{g(x)} \hspace{3 mm} \text{is finit... | https://mathoverflow.net/questions/215798/a-linear-order-from-ap-calculus | [
"co.combinatorics",
"ct.category-theory",
"real-analysis",
"order-theory",
"linear-orders"
] | 19 | 2015-08-27T09:12:46 | [
"This is essentially just a question involving Landau notation, no?",
"With Anthony Quas, I suggest you look up \"Hardy field\", and also \"o-minimal structure\" whose germs at infinity produce Hardy fields. The question as it stands suffers from problems noted by Eric Wofsey.",
"@DmitryV: That's not what I was... | 7 | Science | 0 |
181 | mathoverflow | A question in Fontaine--Laffaille theory | Let $K$ be finite unramified extension of $\mathbf{Q}_p$ with ring of integers $W$. Let ${\rm MF}$ be the category of strongly divisible $W$-modules $M$ with ${\rm Fil}^0M=M$ and ${\rm Fil}^{p-1}M=0$. Let ${\rm Crys}(G_K)$ be the category of finite free $\mathbf{Z_p}$-representations of $G_K$ which become crystalline o... | https://mathoverflow.net/questions/60134/a-question-in-fontaine-laffaille-theory | [
"nt.number-theory",
"galois-representations"
] | 19 | 2011-03-30T15:31:30 | [] | 0 | Science | 0 |
182 | mathoverflow | The oriented homeomorphism problem for Haken 3-manifolds | Haken famously described an algorithm to solve the homeomorphism problem for [the 3-manifolds that bear his name](http://en.wikipedia.org/wiki/Haken_manifold) (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it's often natural to consider 3-manifolds _equipped with an orientation_ , a... | https://mathoverflow.net/questions/107573/the-oriented-homeomorphism-problem-for-haken-3-manifolds | [
"3-manifolds",
"gt.geometric-topology",
"reference-request",
"open-problems"
] | 19 | 2012-09-19T07:37:20 | [
"That's a very nice observation, Ian!",
"Certain special cases can be deduced from the literature. If you look at Theorem 6.1.6 in Matveev's book, it says that there is an algorithm to tell if two Haken manifolds with boundary pattern are homeomorphic taking boundary pattern to boundary pattern. If you have two o... | 4 | Science | 0 |
183 | mathoverflow | Reference request: Parallel processor theorem of William Thurston | Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic automata at each node. There is no such published paper on MathSciNet, and I have no record of this theorem other than ... | https://mathoverflow.net/questions/213181/reference-request-parallel-processor-theorem-of-william-thurston | [
"reference-request",
"graph-theory",
"computational-complexity",
"computer-science",
"geometric-group-theory"
] | 19 | 2015-08-06T08:29:11 | [
"Maybe you could try theoretical computer science, cstheory.stackexchange.com",
"@GerryMyerson: No, it's not in that paper, although the issues of mesh partitions discussed in that paper remind me of the \"bottlenecking\" issues that are at the heart of this parallel processing model I'm asking about, and this m... | 8 | Science | 0 |
184 | mathoverflow | Coarse moduli spaces of stacks for which every atlas is a scheme | Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$. Since $X$ has finite inertia, the coarse space $X^c$ of $X$ exists as an algebraic space.
At _atlas_ of... | https://mathoverflow.net/questions/186379/coarse-moduli-spaces-of-stacks-for-which-every-atlas-is-a-scheme | [
"ag.algebraic-geometry",
"complex-geometry",
"stacks",
"algebraic-stacks",
"algebraic-spaces"
] | 19 | 2014-11-06T08:42:34 | [
"Yes it looks like the example from Corollary 6 does the job! After taking a quick look at the proof, it seems like the condition that the coarse space is a scheme should be equivalent to something like the following: for each point $x \\in P$, there exists a linearized ample line bundle $L$ such that $x$ is $L$-se... | 3 | Science | 0 |
185 | mathoverflow | The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$? | For a partially ordered set $P$, a set $A\subseteq P$ is _cofinal_ if for each element of $P$ there is a larger element in $A$. The _cofinality_ of $P$, ${\rm cof}(P)$, is the minimal cardinality of a cofinal family in $P$.
Let $\kappa$ be an infinite cardinal number. Consider the set $\mathbb{N}^\kappa$ of all functi... | https://mathoverflow.net/questions/221236/the-cofinality-of-mathbbn-kappa-le-for-uncountable-kappa | [
"set-theory",
"gn.general-topology",
"topological-groups",
"foundations"
] | 19 | 2015-10-18T10:35:04 | [
"The Komjath-Shelah model does not involve large cardinals. Just force to add $\\aleph_{\\omega_1}$ many Cohen reals over a model of $2^{\\aleph_0}=\\aleph_1, 2^{\\aleph_1}=\\aleph_2, 2^{\\aleph_2}=\\aleph_{\\omega_1+1}$ and $(\\aleph_{\\omega_1})^{\\aleph_0}=\\aleph_{\\omega_1}$. E.g., force with $Add(\\aleph_2,\\... | 10 | Science | 0 |
186 | mathoverflow | Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian | The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ that associates to a scheme locally of finite type the underlying topological space of its analytification ([SGA1, XII,... | https://mathoverflow.net/questions/364818/homotopy-type-of-the-affine-grassmannian-and-of-the-beilinson-drinfeld-grassmann | [
"ag.algebraic-geometry",
"at.algebraic-topology",
"homotopy-theory",
"geometric-representation-theory"
] | 19 | 2020-07-04T04:17:06 | [] | 0 | Science | 0 |
187 | mathoverflow | support of the coupling between two probability measures | Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the marginal distribution of the first and second coordinate are $\mu$ and $\nu$ respectively. Can we describe the set of all poss... | https://mathoverflow.net/questions/56968/support-of-the-coupling-between-two-probability-measures | [
"pr.probability"
] | 19 | 2011-02-28T21:29:35 | [] | 0 | Science | 0 |
188 | mathoverflow | Does this variant of a theorem of Hasse (really due to Gauss) have an "elementary" proof? | BACKGROUND
Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $S(g)$ consist of the primes, $p$, for which the coefficient of $x^p$ in $g$ is 1. Note that each $p$ in $S(f^k)$ is congru... | https://mathoverflow.net/questions/106267/does-this-variant-of-a-theorem-of-hasse-really-due-to-gauss-have-an-elementar | [
"nt.number-theory",
"quadratic-forms",
"modular-forms",
"power-series"
] | 19 | 2012-09-03T12:27:05 | [] | 0 | Science | 0 |
189 | mathoverflow | Is there some way to see a Hilbert space as a C-enriched category? | The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study metric and then follow Lawvere in considering it a $[0,\infty)$-enriched category; it's particularly nice because the distanc... | https://mathoverflow.net/questions/47644/is-there-some-way-to-see-a-hilbert-space-as-a-c-enriched-category | [
"hilbert-spaces",
"ct.category-theory",
"enriched-category-theory"
] | 19 | 2010-11-28T21:30:13 | [
"You might also be interested in John Baez's paper on the subject: arxiv.org/abs/q-alg/9609018",
"Related question: mathoverflow.net/questions/476/…"
] | 2 | Science | 0 |
190 | mathoverflow | are there high-dimensional knots with non-trivial normal bundle? | Does there exist a smooth embedding $\varphi\colon S^k\to S^n$ such that $\varphi(S^k)$ has non-trivial normal bundle? I looked at some of the old papers by Kervaire, Haefliger, Massey, Levine but I couldn't find an answer. There are various related results, for example it is shown by Kervaire et al that in many dimens... | https://mathoverflow.net/questions/382047/are-there-high-dimensional-knots-with-non-trivial-normal-bundle | [
"at.algebraic-topology",
"gt.geometric-topology"
] | 19 | 2021-01-24T03:39:42 | [
"@StefanFriedl Thanks for the corrections.",
"@archipelago thanks, you put me on the right track!",
"Namely Danny Ruberman writes: \"Jerry Levine's paper \"A classification of differentiable knots\" (Annals 82 (1965) 15-50) determines (see proposition 6.2) the possible normal bundles in some range of codimens... | 13 | Science | 0 |
191 | mathoverflow | About the equivariant analogue of $G_n/O_n$ | Let $BO_n$ and $BG_n$ be the classifying spaces for rank $n$ vector bundles and for spherical fibrations with fiber $S^{n-1}$, respectively, and let $G_n/O_n$ be the homotopy fiber of $BO_n\to BG_n$.
There is an interesting old fact, that the stabilization map $G_n/O_n\to G_{n+1}/O_{n+1}$ is $(2n-4)$-connected, about ... | https://mathoverflow.net/questions/425108/about-the-equivariant-analogue-of-g-n-o-n | [
"at.algebraic-topology",
"homotopy-theory"
] | 18 | 2022-06-20T06:53:22 | [] | 0 | Science | 0 |
192 | mathoverflow | The free complete lattice on three generators, beyond ZF | _This was originally[asked at MSE](https://math.stackexchange.com/q/4261226/28111); although it is still under bounty it seems unlikely to be answered there._
[$\mathsf{ZF}$ proves that there is no free complete lattice on three generators](https://www.jstor.org/stable/1993166?seq=1#metadata_info_tab_contents) since a... | https://mathoverflow.net/questions/405269/the-free-complete-lattice-on-three-generators-beyond-zf | [
"set-theory",
"lo.logic",
"universal-algebra",
"foundations",
"new-foundations"
] | 18 | 2021-10-01T10:56:33 | [
"Given the results in the Crawley-Dean paper, I suspect that if the free complete lattice with $3$ generators exists in a model of $\\mathsf{NFU}$ with choice, then it would actually have the same with any number of generators, including $|V|$.",
"@JamesHanson In fact, I can't even show that \"the free complete l... | 4 | Science | 0 |
193 | mathoverflow | Automorphic forms and coherent cohomology | Why is it (and what does it mean) that automorphic forms do not contribute in the coherent cohomology of Siegel modular varieties parametrizing abelian varieties of dimension $d>2$ (see section 7 of the [thesis](https://tel.archives-ouvertes.fr/tel-02940906/document) of Nguyen or section 1.4.1 of this [paper](https://a... | https://mathoverflow.net/questions/375826/automorphic-forms-and-coherent-cohomology | [
"nt.number-theory",
"arithmetic-geometry",
"automorphic-forms",
"galois-representations"
] | 18 | 2020-11-06T10:32:18 | [
"A good reference for coherent cohomology of Shimura varieties is the paper of Michael Harris, cited as [Har90b] in the BCGP paper. In particular, Proposition 4.3.2 there gives a formula for the infinitesimal character associated to the coherent cohomology of an automorphic vector bundle. You might also find the pa... | 10 | Science | 0 |
194 | mathoverflow | Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$ | Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\\}$.
I Conjecture
$$\sum_{n \in A_p } n=\dfrac{7(p^2-1)}{24}$$
The question has already been asked in this [thread](https://math.stackexchange.com/questions/4... | https://mathoverflow.net/questions/440995/mysterious-sum-equal-to-frac7p2-124-where-p-equiv-1-pmod4 | [
"nt.number-theory",
"prime-numbers"
] | 18 | 2023-02-16T06:48:42 | [
"In the same vein, see also in Franz Lemmermeyer's book \"Reciprocity Laws\", especially chapter 1 and exercise 1.32 (Gauss).",
"@SB1729 Good question",
"@Pascal I did some simple computations, this problem is so interesting feels like doing more, so I think we have $\\sum_{a\\in A_p}a^2=\\frac{(p-1)(5p^2+7p-1... | 12 | Science | 0 |
195 | mathoverflow | Two curious series for $1/\pi$ | On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have $$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol and $$R_p:=\frac1p\sum_{n=0}^{p-1}\frac{6n+1}{(-1728)^n}\binom{2n}n\sum_{k=0}^n\binom nk\binom{n+2k}... | https://mathoverflow.net/questions/369569/two-curious-series-for-1-pi | [
"nt.number-theory",
"sequences-and-series",
"combinatorial-identities",
"congruences",
"ramanujan"
] | 18 | 2020-08-19T07:20:37 | [
"Thank you, this is very interesting!",
"You may look at my talk available from maths.nju.edu.cn/~zwsun/CNT-talk3.pdf and my recent paper available from dx.doi.org/doi:10.3934/era.2020070.",
"What is the relation between the congruence and the identity for pi? Is there an explanation for why we would expect suc... | 3 | Science | 0 |
196 | mathoverflow | Cycles in algebraic de Rham cohomology | Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, there is a Chern class map $$c_{1,t}:\text{Pic}(X_t)\to H^2_{dR}(X_t),$$ where $H^2_{dR}$ denotes algebraic de Rham coh... | https://mathoverflow.net/questions/429644/cycles-in-algebraic-de-rham-cohomology | [
"ag.algebraic-geometry",
"arithmetic-geometry",
"algebraic-cycles",
"chern-classes",
"derham-cohomology"
] | 18 | 2022-09-02T07:26:35 | [
"@PeterScholze: Thanks! This does indeed seem quite related, and indeed the discussion on page 81 of \"Une Introduction aux Motifs\" is closely related to the argument François Charles sketched for the case of Abelian varieties.",
"Related in spirit, but I think a bit different, is the Ogus conjecture, discussed ... | 7 | Science | 0 |
197 | mathoverflow | Čech functions and the axiom of choice | A **Čech closure function** on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$, and (iii) $\varphi(X\cup Y)=\varphi(X)\cup\varphi(Y)$ for all $X,Y\subseteq\omega$; in other words, it obeys the K... | https://mathoverflow.net/questions/361038/%c4%8cech-functions-and-the-axiom-of-choice | [
"gn.general-topology",
"set-theory",
"axiom-of-choice"
] | 18 | 2020-05-20T03:03:19 | [
"@Haim Looks to me like they both use choice. My guess is, if you can construct such a function without choice, it will be by a completely different method.",
"@bof Out of Theorem 1 and Theorem 2 in the Galvin-Simon paper, which one is using AC in its proof?",
"@YCor: I got a partial answer to this in this ques... | 9 | Science | 0 |
198 | mathoverflow | Is the Frog game solvable in the root of a full binary tree? | _This is a cross-post from[math.stackexchange.com](https://math.stackexchange.com/q/3800570/318073)$^{[1]}$, since the bounty there didn't lead to any new insights._
* * *
For reference,
> The **Frog game** is the generalization of the _Frog Jumping (see it on[Numberphile](https://www.youtube.com/watch?v=X3HDnrehyDM... | https://mathoverflow.net/questions/370694/is-the-frog-game-solvable-in-the-root-of-a-full-binary-tree | [
"graph-theory",
"trees",
"induction",
"binary-tree"
] | 18 | 2020-09-02T09:53:11 | [] | 0 | Science | 0 |
199 | mathoverflow | Orientation-reversing homotopy equivalence | If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there is an orientation-reversing homeomorphism there must be an orientation-reversing diffeomorphism (consider exotic 7-sphe... | https://mathoverflow.net/questions/373348/orientation-reversing-homotopy-equivalence | [
"dg.differential-geometry",
"at.algebraic-topology",
"gt.geometric-topology",
"homotopy-theory"
] | 18 | 2020-10-05T06:15:17 | [
"This can be heuristically reduced to finding manifolds that are homotopy equivalent but not homeomorphic, which should show you how to organize thinking about this. If $X$ does not admit an orientation-reversing homotopy equivalence and $Y$ is homotopy equivalent to $X$ but not homeomorphic, then the connected sum... | 6 | Science | 0 |
200 | mathoverflow | Are these local systems on $\mathscr{M}_{g,1}$ motivic? | Let $(\Sigma_g, x)$ be a pointed topological surface of genus $g$, and let $MCG(g,1)$ be the mapping class group of this pointed surface. Then $MCG(g,1)$ has a natural action on $\pi_1(\Sigma_g, x)$ $$MCG(g)\to \text{Aut}(\pi_1(\Sigma_g))$$ and hence acts on the set of irreducible characters of $\pi_1(\Sigma_g)$: $$\ga... | https://mathoverflow.net/questions/345955/are-these-local-systems-on-mathscrm-g-1-motivic | [
"ag.algebraic-geometry",
"gt.geometric-topology",
"mapping-class-groups"
] | 18 | 2019-11-13T08:17:12 | [
"@BenWieland: (1) is an interesting idea -- I'll play with it a bit and see what I get. Thanks!",
"Oops, my (2) is not reducible as an $M_{g,1}$ representation. That was the whole point, yet it ruins it. The whole point is that it is the degeneration of an extension to a split extension, but such degenerations ar... | 12 | Science | 0 |
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