All Questions
5,076 questions with no upvoted or accepted answers
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A duality on partial permutations
A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get ...
- 27.9k
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427
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Non-commutative Formal Group Laws
Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of ...
- 10.5k
9
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3k
views
"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
9
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0
answers
2k
views
Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
- 20.5k
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644
views
Conceptual proofs for the computation of the structure sheaf
The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...
- 63.1k
9
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241
views
An inequality for the ratio of standard Young tableau with {1,2,...,k} in the first row
For a partition $\lambda \vdash n$, define $\dim \lambda$ to be the number of standard Young tableaux of shape $\lambda$, and $\dim \lambda/(k)$ as the number of standard Young tableaux with $1,2,\...
- 181
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391
views
Intersection of plus/minus cells in Bialynicki-Birula decomposition
Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:
$X = \bigsqcup_{w\in W} C_w$ ...
- 1,595
9
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885
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Continuous projections in $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $...
- 1,697
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543
views
Maximum volume convex body coverable by a unit square
Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...
- 151k
9
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329
views
'Infinitesimal' elements of a topological group
Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...
- 4,738
9
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685
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Name for a topological space where every closed set contains a closed point
A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...
- 1,822
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205
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Reference for sparseness of incomparability graphs implying sparseness of covering graphs
If a partial order on $n$ elements has $m$ incomparable pairs, then its covering graph (aka Hasse diagram aka transitive reduction, the graph of pairs of elements that are comparable but are not the ...
- 18.6k
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462
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$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$
Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
- 3,076
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560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
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605
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Hilbert symbol and Weil index, beyond the quadratic case?
Let $F$ be a local nonarchimedean field. Let $n$ be a positive integer for which the group $\mu_n(F)$ of $n^{th}$ roots of unity in $F$ has order $n$. Let $\epsilon: \mu_n(F) \rightarrow C^\times$ ...
- 13.3k
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1k
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Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 2,976
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answers
732
views
Kontsevich's lecture at Rutgers (1996)
Reading some papers about (homological) mirror symmetry I have found the reference to the unpublished Kontsevich's lecture at Rutgers University (Nov 11, 1996). I would like to know if someone has ...
9
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309
views
Higher-order dimension in posets: a reference request
Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...
- 4,462
8
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245
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Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,462
8
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answers
150
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What is the forcing $\bf U$ from Bartoszyński-Judah?
In Set Theory - on the structure of the Real Line by Bartoszyński & Judah, a forcing notion $\bf U$ is mentioned on page 339, allegedly corresponding to $\rm{cof}(\cal N)$ as it has several ...
- 403
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116
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optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
8
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208
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Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
8
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answers
156
views
Square root of an Anosov diffeomorphism
Let $T\colon \mathbb T^d\to\mathbb T^d$ be an Anosov diffeomorphism (that is, the tangent bundle splits into invariant stable and unstable bundles; the restriction of $DT$ to the unstable bundle is ...
- 23.2k
8
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341
views
Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 773
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370
views
An obscure case of Curry-Howard
It is a theorem of the Intuitionistic Propositional Calculus that
$$
(p\to q)\to p = (q\to p) \land ((p\to q)\to q).
$$
The Curry-Howard correspondence realizes this as a pair of operators (for any ...
- 18.4k
8
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answers
221
views
Elkies' family of elliptic curves of rank 19
There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list
at [email protected]&...
- 161
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answers
197
views
Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
- 1,155
8
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answers
156
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Generators for unstable cobordism
I am looking for explicit descriptions of generators of some low-dimensional unstable cobordism groups. For example, $\mathbb CP^2$ embeds into $\mathbb R^7$ by a result of Haefliger. Because it has ...
- 6,813
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answers
198
views
Subleading terms in Weyl's Law
The two term Weyl's conjecture states that
$$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$
where $\Omega$ is a ...
- 81
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answers
138
views
Relation between "homotopical" and "representation-theoretic" categorifications
This might be a bit of a soft question, and I apologize in advance for this. Here it is:
What is the relationship between the "homotopical" categorification (e.g. we consider every category ...
- 1,374
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190
views
A reference on a result by Steve Schanuel
In the Author Commentary section of the TAC reprint of the paper of 1968 Diagonal arguments and cartesian closed categories., Bill Lawvere wrote:
‘Nilpotent infinitesimals fall far short of even one-...
- 774
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202
views
Looking for the source of expository passages concerning the Adams spectral sequence
The following question might be off-topic strictly speaking, but it does have the form of a reference request (in some sense) and also I think that enough members of the relevant community are MO ...
- 25.8k
8
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245
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Did these graphs pop up somewhere?
Please let me know if the following graphs popped up in some problems.
Each of these graphs is described by 5 integers $n_1\geqslant k_1$, $n_2\geqslant k_2$, $l\geqslant 0$.
We take two complete ...
- 45k
8
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answers
484
views
Analytification of DG-categories over $\mathbb C$?
In recent notes of complex geometry by Clausen–Scholze, they gave a theory of analytification of finite type $\mathbb C$-schemes. It seems to me that there is a non-commutative analogue which works ...
- 2,866
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votes
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answers
137
views
Group presentations where discarding generators always yields a subgroup
Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\...
- 1,005
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answers
112
views
Identity for the associator involving a third root of unity
This is a reference request. I came across the class of nonassociative algebras satisfying the following identity:
$$
(a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0.
$$
Here:
by an "algebra" I mean a ...
- 16.9k
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0
answers
240
views
Linear $S^{2k}$-bundles over $S^{4k}$
By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...
- 291
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441
views
Poincaré duality for topological $K$-theory
Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with
$H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$.
$H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user39380
8
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answers
235
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Stability of nodal hypersurfaces
We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...
8
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220
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Key ideas behind p-adic Baker's theorem
I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
- 459
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answers
695
views
In need of help with parsing non-Archimedean function theory
My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...
- 1,284
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311
views
Unions=colimits in categories
The basic way to define a partial map $X\rightharpoonup Y$ in a category is as a span $X\hookleftarrow U\to Y$ in which the first map (the support) is mono and we call the second evaluation. These are ...
- 8,481
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284
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Does Borel fixed-point theorem hold for Deligne-Mumford stacks?
Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus.
Question: Is the following statement true?
...
8
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answers
585
views
Scholze and Weinstein's $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}_p$
In their Berkeley Lectures, to motivate the introduction of Diamonds, Scholze and Weinstein discuss what should be the definition of $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}...
- 1,479
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answers
367
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References for Yoichi Miyaoka's work around Fermat's Last Theorem
Apparently, Yoichi Miyaoka made a serious attempt to prove FLT in 1988. See the following question.
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last ...
8
votes
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answers
304
views
On the remainder of a power series evaluated on the boundary of its convergence disk
Background
This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
- 6,377
8
votes
0
answers
299
views
(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request
Suppose we have a tower of Kan fibrations between Kan complexes:
$$ X_0 \xleftarrow{f_0} X_1 \xleftarrow{f_1} X_2 \xleftarrow{f_2} \dotsb $$
From this we get a commutative diagram of topological ...
- 56.9k
8
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answers
165
views
Original reference for the Fam construction
For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...
- 10.7k
8
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646
views
Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?
I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word ...
- 18.4k
8
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answers
300
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356