Unanswered Questions
49,209 questions with no upvoted or accepted answers
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Surprisingly only real points on intersection of certains quadrics
Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by
\begin{align}
X_e &= 0\\
X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\
X_g &...
- 22.5k
14
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318
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Do Sullivan's and Wilkerson's algebraic group structures on rational homotopy automorphisms coincide?
For a 1-connected space $X$ with the homotopy type of a finite CW-complex, the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X)$ are known to have the structure of an arithmetic group up to finite ...
- 8,167
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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
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831
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Growth of residues of $1/\zeta(s)$: conjectures?
Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...
- 20.2k
14
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821
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What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
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480
views
How should we think about the algebraic moment map?
My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...
- 6,292
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288
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Which topoi are local with respect to Stone-Cech compactification?
Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
- 64k
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388
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Describing the points of a glued topos
Let $f : \mathbf{X}\to \mathbf{Y}$ be a morphism of topoi; in his 1977 monograph, Johnstone describes the open mapping cylinder of $f$ as the following pushout of topoi:
$\require{AMScd}$
\begin{CD}
\...
14
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788
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Covering image of a connected CW-complex need not be a CW-complex
This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
- 632
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358
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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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343
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Do connected algebraic stacks have a smooth cover by a connected scheme?
An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
- 1,104
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420
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Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
- 1,902
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1k
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The quantization problem: modern quantization procedures and current status
The quantization problem is one of the most well-known current problems of theoretical and mathematical physics. It is also part of Hilbert's sixth problem (on the axiomatization of physics - see here ...
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863
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strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
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718
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Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
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358
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What is the asymptotic dynamics of the winning position in this game?
$n$ players indexed $1,2,...,n$ play a game of mock duel. The rules are simple: starting from player $1$, each player takes turns to act in the order $1,2,...,n,1,2,...$. In his turn, a player ...
- 2,619
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851
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Cardinality of the set of continuous functions
Suppose $(X,\tau)$ and $(Y,\sigma)$ are topological spaces. Let $F(X,Y)$ be the set of continuous functions $X\rightarrow Y$. I want to compute the cardinality of $F(X,Y)$. It depends not only on ...
- 12.4k
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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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551
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Seeing what gets Harvey Friedman's "tangible incompleteness" principles into large cardinal territory
I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
- 378
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225
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Hauptvermutung for non-manifolds
The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...
- 2,050
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314
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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"
...
- 747
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262
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
- 2,306
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1k
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Nice proof of inequality $(1-x^p)^{1/p}(1-x^q)^{1/q}\ge (1-x)(1+x^c)^{1/c}$ where $2^{1/c} = p^{1/p} q^{1/q}$?
Let $0\leq x < 1$, $1 \leq p < \infty$ and $q$ be the conjugate exponent defined by
$$1/p + 1/q = 1.$$
I am looking for a nice proof that
$$ \frac{(1-x^p)^{1/p}(1-x^q)^{1/q}}{(1-x)(1+x^c)^{1/...
- 2,283
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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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500
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The Ax-Kochen isomorphism theorem and the continuum hypothesis
Let's recall that:
(1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...
- 32.2k
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884
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On mixed $p$-adic Hodge theory
Does mixed $p$-adic Hodge theory exist? Can we extend the scope of comparison theorems using simplicial resolutions a la Deligne? Do we get 3 opposite filtrations as in classical mixed Hodge theory, ...
- 141
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297
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Ordinal-valued sheaves as internal ordinals
Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
- 32.5k
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378
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When is a map of topological spaces homotopy equivalent to an algebraic map?
My question is simple, but I don't expect there are any simple answers.
Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
- 1,635
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336
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Are there exotic twisted doubles of 4-manifolds?
Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
- 515
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654
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Reverse Mathematics of Euclid's theorem
Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
- 35.5k
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321
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The "concreteness preorder" on categories
A category is called concrete if it is equipped with faithful functor to $\mathrm{Set}$. Let us say a category $C$ can be made concrete if there exists a faithful functor $F \colon C \to \mathrm{Set}$...
- 22.3k
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558
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Rings that fail to satisfy the strong rank condition
In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...
- 282
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405
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O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
- 32.2k
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988
views
Witten zeta function v.s. Riemann zeta function
From a talk, we learned that
The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”:
where we sum over irreducible ...
- 10.5k
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225
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Open questions about automorphism tower theorem
Joel Hamkins has left four open questions about automorphism tower theorem in his wonderful paper Every group has a terminating transfinite automorphism tower.
In fact, the four questions are "...
- 365
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811
views
Cardinality vs. isomorphism type of vector spaces without choice
One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem:
If $V$ is an infinite vector space over a field $F$, and $...
- 39.8k
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245
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How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$
Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...
- 549
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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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1k
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Explicit example of elliptic curve of the kind needed for IUTT
At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
- 173
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546
views
To what extent does Floer cohomology detect Hamiltonian non-displaceability of immersed curves?
Floer cohomology for immersed Lagrangians is introduced by
Akahi, Manabu; Joyce, Dominic, Immersed Lagrangian Floer theory, J. Differ. Geom. 86, No. 3, 381-500 (2010) and its one-dimension version (...
- 1,628
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932
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Relation between Igusa tower and $p$-adic modular forms
As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
- 445
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716
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Algebra for the Baby
I am reading the following article.
Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012..
Author works with 4370-...
user21230
14
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543
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Small cardinals related to topological convergence
Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
- 4,413
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353
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The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
- 32.2k
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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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530
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Knots with high unknotting number relative to their genus
Can the unknotting number of a knot with fixed three-genus be arbitrarily high?
Some background and motivation: the unknotting number $u(K)$ of a knot $K$ (i.e. the minimum number of necessary ...
- 890
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1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
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338
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Are there Alexander-Whitney maps in geometric homology?
When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology -
let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-...
- 9,580
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205
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Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
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919
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Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,129