Unanswered Questions
49,207 questions with no upvoted or accepted answers
16
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Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
- 20.5k
16
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0
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325
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Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
- 23.5k
16
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734
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What is the current status of the question of whether or not the mapping class group has Kazhdan's Property (T)?
$\DeclareMathOperator\Mod{Mod}$Let $\Mod(S)$ be the mapping class group of a closed oriented surface $S$ of genus at least $3$. My question is easy to state: is it currently known whether or not $\...
- 161
16
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369
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Transcendence of sum of reciprocals of factorials
For $A \subseteq \mathbb{N}$, define $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$. It is easy to see that for every infinite $A$, $x_A$ is irrational.
Question: Is there an infinite $A \subseteq \...
- 161
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558
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Identity involving Schur polynomials, binomial coefficients and contents of partition
Let $C_{\lambda,\mu}$ be the coefficients defined as
$$ s_\lambda\left(\frac{x_1}{1-x_1},...,\frac{x_N}{1-x_N}\right)=\sum_{\mu\supset \lambda}C_{\lambda\mu}s_\mu(x_1,...,x_N),$$
where $s$ are the ...
- 2,552
16
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288
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Fibrations whose total spaces are more highly connected than their fibers
The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers.
Are ...
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16
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222
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Reference request: Milnor rank of spheres
Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
- 343
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875
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"Geometric" proof of Kunneth formula
The usual proof of the Kunneth formula (say for either the homology or cohomology of manifolds) is essentially pure homological algebra. I was wondering if there was a more geometric proof, i.e., one ...
- 1,046
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437
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Can non-reduced fibers appear over a subset of codimension $\geq 2$?
I already asked this on math.stackexchange.com, but didn't receive an answer.
Suppose $f: X \to Y$ is a (possibly proper) morphism of complex manifolds (resp. smooth varieties) such that all fibers of ...
- 1,286
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755
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Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 18.4k
16
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475
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Progess on a Problem/Conjecture of Sullivan?
In Sullivan's postscript to his MIT notes https://www.maths.ed.ac.uk/~v1ranick/surgery/gtop.pdf he describes some problems and conjectures, where Problem 4 is: "Analyze the action of Gal($\...
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Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,746
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400
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Quadratic non-residues in elliptic divisibility sequences
Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in ...
- 21.4k
16
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787
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Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,z\in\mathbb Z$
Roger Heath-Brown conjectured that any integer $k\not\equiv\pm4\pmod9$ can be expressed as $x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$ in infinitely many different ways. Also it is well-known that some ...
- 701
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1
answer
688
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Approximating zero sets of real polynomials with "less complicated" polynomials
Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
- 1,407
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574
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Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
- 1,206
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631
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The Octahedral Axiom in group theory
$\require{AMScd}$Here are two results about groups:
(The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$. ...
- 156k
16
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213
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Characterization of geometric morphisms without referring explicitly to the left adjoint?
Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
- 64k
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780
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Shimura varieties and connected components
Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\...
- 37k
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372
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On projectively countable sets in the Hilbert cube
A subset $A$ of a topological space $X$ is called projectively countable if for any continuous map $f:X\to\mathbb R$ the image $f(A)$ is countable.
It is easy to see that each projectively countable ...
- 42k
16
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556
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Reference request for Grothendieck's work on "Integration with values in a topological group"
Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
user57432
16
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0
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646
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Consistency strength of $j:L_δ→L_δ$ for some δ
What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$?
The consistency strength is strictly between totally ineffable and $ω$-Erdős ...
- 7,545
16
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489
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An inequality for matrix norms
Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:
Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
- 4,484
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952
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Continuous cohomology of a profinite group is not a delta functor
Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...
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255
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Generalization of Newton's identities to Schur functions
In some recent work, I've stumbled across the following identity for $\lambda \vdash n$:
$$
n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu.
$$
Here, ...
- 1,989
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345
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What property do small primes have that prevent the existence of a Tarski monster?
For an odd prime $p$, a Tarski monster group is an infinite group $G$ such that every proper, non-trivial subgroup $H < G$ is a cyclic group of order $p$. It is known that for every prime $p > ...
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411
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Simple disproof of Danzer — Grünbaum conjecture
I asked this question on the MSE, but I did not get an answer. I hope that one of the experienced participants will check the correctness of the proof or the truth of the statement (and, perhaps, will ...
- 269
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What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
- 18.4k
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589
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The number 1680 and Lagrange's four-square theorem
The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...
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315
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A question on surfaces in $\mathbb{P}^4$
On surfaces in $\mathbb P^4$,Ellingsrud and Peskine has proved that
There exists an integer $d_0$ such that for any integer $d>d_0$,any smooth surface of degree $d$ in $\mathbb P^4$ is of ...
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616
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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}...
- 28k
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255
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Abelian varieties (over $\mathbb{Q}$) with large Mordell-Weil rank
Let $A$ be an abelian variety defined over $\mathbb{Q}$ of dimension $g \geq 1$. We shall denote by $A(\mathbb{Q})$ the Mordell-Weil group of rational points in $A$, and denote by $r = r_A$ the rank ...
- 26.9k
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424
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Do TQFTs give a complete set of invariants of manifolds?
An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by ...
- 1,430
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438
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Relationship between the twistor spaces due to Penrose and Salamon
In Penrose's construction of the twistor space of Minkowski spacetime $\mathbb R^{1,3}$, we first complexify $\mathbb R^{1,3}$ to $\mathbb C^4$ and then think of points in it as matrices acting on $\...
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Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type
Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...
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Kodaira-Spencer maps and deformation theory
This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures?
The deformation theory of ...
- 3,020
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603
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K-theory and homology of groups
It is known that for any ring $R$,
$$K_{1}(R)=H_{1}(GL_{\infty}(R), \mathbb{Z})$$
$$ K_{2}(R)= H_{2}(E_{\infty}(R),\mathbb{Z})$$
$$ K_{3}(R)= H_{3}(St_{\infty}(R),\mathbb{Z})$$
where $GL_{\infty}= ...
- 1,613
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247
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Gap two Sierpinski set?
Is it consistent to have a set of reals $X$ of size $\aleph_3$ such that for every $Y \subseteq X$, $Y$ has measure zero iff $|Y| \leq \aleph_1$?
- 9,641
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644
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Dijkgraaf-Witten topological invariant
We know that given a finite group $G$ and its group 4-cohomology class $w \in H^4[G;U(1)]$, we can obtain a DW topological invariant $Z_{G,w}(M^4)$ as the partition function of the DW theory on a ...
- 4,826
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591
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Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?
Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...
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382
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Representation categories and homology
Let $G$ be a finite group.
Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$.
Can we recover some homological ...
- 5,039
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717
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Decidable open problems
Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or ...
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Connected sum is well-defined for surfaces, proof?
EDIT: So my question is distinct from the question asked here because I am asking an easier question. Why should we have to invoke something as powerful as the "Annulus Theorem" to show that the ...
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363
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Combinatorial characterization of intersecting intervals in the plane
Consider $n$ points $A=\{A_1,\dotsc,A_n\}$, and another set of points, $B=\{B_1,\dotsc,B_n\}$ in the plane. We can assume they are all disjoint.
For each permutation $\pi$, consider the collection of ...
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396
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Division of a square and value of a disk
[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk
I cam across this problem ...
- 477
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988
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A Combinatorial Game: the Snake and the Hunter
The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...
- 3,707
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1
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743
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Inequalities for marginals of distribution on hyperplane
Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
- 156k
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439
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Survey of known results on equivariant transversality
Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant ...
- 9,580
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366
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Specific cases of the tangle hypothesis in terms of "classical" n-categories
As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-...
- 863
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0
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531
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Aligned roots of irreducible polynomials
It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
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