All Questions
5,101 questions with no upvoted or accepted answers
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Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
15
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454
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Grothendieck dessins d'enfants - current surveys or text you can recommend?
I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
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330
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How much smoothness does the tennis ball theorem need?
The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are ...
- 18.6k
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602
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Precise form of the mean motion theorem
Consider an exponential polynomial
$$f(t)=\sum_{k=1}^na_k\exp(i\lambda_kt),$$
where $a_k$ are complex and $\lambda_k, t$ real. The usual form of the Mean Motion Theorem says that the limit
$$\lim_{t\...
- 91.8k
15
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404
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Extending Kahler metric from a neighborhood of a divisor to the whole manifold
Let $X$ be a smooth complex projective variety with an ample line bundle $L$, and let $D\subset X$ be a smooth divisor. Suppose in an analytic neighborhood $U$ of $D$ there is a Kahler form $\omega$ ...
- 14.3k
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397
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References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
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576
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Geometry underlying a comparison of Dieudonné theories
Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$.
There are several presentations of the ...
- 6,308
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3k
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Weak$^*$ convergence of measures vs. convergence of supports
Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
- 1,658
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716
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Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
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357
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Existence of flat connections via characteristic classes, for nice groups
I have two questions about what I write below (which honestly seems pretty elementary).
Is it true (more or less)?
Is there a clean reference that I can cite.
Let $G$ be a compact Lie group, $M$ a ...
- 27.2k
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448
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Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?
I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...
- 43.2k
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382
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Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?
In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' $H_{dR}^1(Z,\mu)...
user14166
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477
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Expanding disks lead to what packing of the plane?
Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...
- 151k
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573
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Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane
This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...
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591
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For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
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Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
- 18.7k
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476
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Any references on zeta-function like sums of inverse determinants over lattices of matrices?
I'm sorry for the title, it was little difficult to phrase..
Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...
- 201
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885
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How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?
This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 (...
- 52.9k
14
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362
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Comparing algebraic and analytic spaces through the universal property of classifying topoi
$\newcommand\kAlg{k\mathrm{Alg}}\DeclareMathOperator\Zar{Zar}\newcommand\Mnf{\mathrm{Mnf}}$I apologize beforehand if my question is naïve. I must admit that I do not know much about analytic/smooth ...
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298
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Representation theory of Kac-Moody algebras in positive characteristic
I have been trying to learn about the representation theory of Kac-Moody algebras in positive characteristic. However, my usual reference for the subject (Infinite dimensional Lie algebras by Kac) ...
- 1,389
14
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357
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How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
- 3,539
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404
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Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
- 208
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312
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An unpublished paper of Thurston about diffeomorphism groups
William Thurston has done many contributions in the field of diffeomorphism groups. But it seems that one of his papers entitled
"On the Structure of the Group of Volume Preserving Diffeomorphisms"
...
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1k
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Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
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481
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If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 109k
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270
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A symmetry of lattice paths
The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity.
...
- 2,896
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618
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Chasing a 1950s thesis from the University of Dhaka on block designs
On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below.
Author: Qazi Motahar Husein (Sometimes Husain or Hussein).
Title of the Thesis: Symmetrical ...
- 6,162
14
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644
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Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?
Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
- 13.8k
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3k
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Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
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552
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Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.7k
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522
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Reconstruction conjecture and partial 2-trees
Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old.
Searching relevant literature,...
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Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
13
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331
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Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
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2k
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Is there any correspondence between Gödel and Kreisel that supports Kreisel's observation that Gödel changed his mind about his 1938 set theory note?
At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in ...
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328
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Upper bound on prime powers in interval
I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses:
(a) the ...
- 20.2k
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325
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$\mathsf{AC}_\mathsf{WO}+\mathsf{AC}^\mathsf{WO}\Rightarrow \mathsf{AC}$?
Let
$\mathsf{AC}_\mathsf{WO}$: Every well-orderable family of non-empty sets has a choice function.
$\mathsf{AC}^\mathsf{WO}$: Every family of non-empty well-orderable sets has a choice function.
My ...
- 2,286
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364
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What is known about differentiable and analytic structures on the long line (and half-line)?
When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...
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800
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Reference request for a complete and formal Duality Principle in category theory
Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...
- 63.1k
13
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337
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Morava K-theory of loop spaces of spheres
Some time ago I cam across the paper "What we still don't know about loop spaces of spheres" by Ravenel:
https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf
which concerns ...
- 1,759
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A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
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389
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Prime numbers of the form $1+p+p^2+\dots+p^n$
I am looking at prime numbers of the form $Q=1+p+p^2+\dots+p^n$, where $p$ is also a prime number, $n$ is finite. Unfortunately I was unable to find any reference to such prime numbers in the ...
- 243
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Where is a full proof of the precise Torelli theorem in the literature?
The precise form of Torelli's theorem is as follows (translated from Serre's appendix to Lauter - Geometric methods for improving the upper bounds on the number of rational points on algebraic curves ...
- 3,539
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182
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What do we call a functor of orbits and isomorphisms?
If $G$ is a finite group, then inside the category of $G$-sets and $G$-maps there is the subcategory whose objects are the orbits (transitive $G$-sets) and whose morphisms are the isomorphisms. I have ...
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606
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Algebraic closure of a field in constructive mathematics
There is a short note by André Joyal called Les théorèmes de Chevalley-Tarski et remarque sur l'algèbre constructive (pp. 256-258). It is claimed there that there is a constructive version of the ...
- 4,459
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797
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Borel-Weil-Bott, Langlands and Hitchin
Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$.
Given a dominant weight $\lambda$, one can construct a line ...
- 2,273
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323
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Reference request: exponential growth rates of subword-closed languages are integers
For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters (...
- 2,339
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880
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Arguments against Freiling's argument against Continuum Hypothesis
Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
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349
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Some $q-$analogues of $ \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}.$
Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} )$ and
let $ {{n}\brack{k}}_q$ denote a $q-$binomial coefficient.
I am interested in $q-$analogues of the identity $ \sum\...
- 5,612
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523
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Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
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475
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Singular cohomology of $BG$ and Borel cohomology of $G$
Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
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