Unanswered Questions
49,206 questions with no upvoted or accepted answers
29
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989
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Non-linear expanders?
Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
- 20.2k
29
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answers
3k
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Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
29
votes
0
answers
747
views
Why do H_4 and M_4 have the same virtual Euler characteristic?
Here's a funny coincidence:
The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$.
Now ...
- 40.2k
29
votes
0
answers
2k
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Nontrivial tangent bundle that is diffeomorphic to the trivial bundle
Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$?
(This ...
- 10.3k
29
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0
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3k
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
- 3,022
28
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546
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Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Although the question is fairly unambiguous, I give all relevant definitions:
$\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-...
- 7,276
28
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527
views
What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
- 37.8k
28
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0
answers
2k
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How did Gauss find the units of the cubic field $\mathbb Q[n^{1/3}]$?
Recently I read the National Mathematics Magazine article "Bell - Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas ...
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28
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716
views
Does this infinite primes snake-product converge?
This re-asks a question I posed on MSE:
Q. Does this infinite product converge?
$$
\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
- 151k
28
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0
answers
2k
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Is Feferman's unlimited category theory dead?
In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
- 1,446
28
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0
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676
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Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
- 22.5k
28
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0
answers
1k
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Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 6,486
28
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827
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Can one divide by the cardinal of an amorphous set?
This question arose in a discussion with Peter Doyle.
It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X \...
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28
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907
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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 ...
- 6,819
28
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828
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Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...
- 151k
28
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0
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3k
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The Work of Pierre Deligne
In this biography of Pierre Deligne, there is a quote of Jacques Tits which says that "quite a few of his best ideas have never been written!".
What are some of his best ideas that you have heard of ...
28
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0
answers
1k
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Derivative of Class number of real quadratic fields
Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$,
and define the Fekete polynomials
$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$
Define
$$ f_N(X) = \frac{F_N(X)}{...
- 32.6k
28
votes
0
answers
1k
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On the (derived) dual to the James construction.
Background
If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by
$$
X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots
$$
in ...
28
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0
answers
2k
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derived category of equivariant coherent sheaves and fixed points
The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient ...
- 3,051
28
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0
answers
4k
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What's the intuition between formal smoothness, etaleness. and unramifiedness?
Let $f: X \to Y$ be a morphism of schemes. Then $f$ is called (EGA IV.17) formally smooth if whenever $T$ is an affine $Y$-scheme and $T'$ a closed subscheme of $T$ defined by a nilpotent ideal (it's ...
- 25.6k
28
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0
answers
2k
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Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
- 25.5k
28
votes
1
answer
2k
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Analogy between the exterior power and the power set
The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...
- 63.1k
27
votes
0
answers
469
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,020
27
votes
1
answer
1k
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The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
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0
answers
960
views
Nearby cycles without a function
Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...
- 6,162
27
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0
answers
1k
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Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
27
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0
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940
views
A question on simultaneous conjugation of permutations
Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the ...
- 271
27
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0
answers
1k
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Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 366
27
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0
answers
1k
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Computational complexity of topological K-theory
I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
- 666
27
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0
answers
2k
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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 $\...
- 20.5k
27
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answers
3k
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26
votes
0
answers
514
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A non-self-intersecting unit side length polygon in a unit square has odd number of sides unless it is the square itself
This is the same question as here in SE.
I have a conjecture, it is like this:
Suppose there is a non-self-intersecting polygon lies inside a closed square of length $1$. The polygon has every side ...
- 843
26
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0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
26
votes
0
answers
359
views
Can 4-space be partitioned into Klein bottles?
It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...
- 151k
26
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0
answers
1k
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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 ...
- 156k
26
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0
answers
908
views
Where to submit this work with several unusual features?
I appreciate that questions about where to submit are generally considered off-topic, but I hope that the unusual features of the present case may make it acceptable.
I have put a monograph on github ...
- 56.9k
26
votes
0
answers
841
views
Is the Flajolet-Martin constant irrational? Is it transcendental?
Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...
- 3,028
26
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0
answers
1k
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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 ...
- 20.5k
26
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0
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910
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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 ...
- 1,513
26
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0
answers
708
views
Coloring a Ferrers diagram
I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed.
Say that a coloring of the dots of a Ferrers diagram ...
- 82.7k
26
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0
answers
642
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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 ...
- 865
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0
answers
1k
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The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)
I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor (http://www.ams.org/journals/jams/2012-25-03/S0894-0347-...
- 16.9k
26
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0
answers
2k
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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 ...
- 1,362
26
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0
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657
views
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 ...
- 1,875
25
votes
0
answers
2k
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Is there a proof of Hodge theory using condensed mathematics?
As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i....
- 761
25
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0
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619
views
Galois representations attached to Shimura varieties - after a decade
In an answer to the question Tools for the Langlands Program?, Emerton, in his usual illuminating manner, remarks on the reciprocity aspect of Langlands Program: "...As to constructing Galois ...
- 1,267
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1k
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What, precisely, do we mean when we say that a f.d. vector space is canonically isomorphic to its double dual?
I've been reading the Xena Project blog, which has been loads of fun. In the linked post Kevin gives the natural isomorphism $V \to V^{\ast \ast}$ from a f.d. vector space to its dual as an example of ...
- 118k
25
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377
views
Reference request for educational material In source format, for blind accessibility purposes
Introduction
I am a blind undergraduate studen in mathematics. I use screen reading software, which uses synthesized speech to read aloud the contents of the screen, to read and write math.
Due to ...
- 353
25
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0
answers
1k
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Caramello's theory: applications
In this 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 ...
user140765
25
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0
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752
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What is a Green's function in the language of $\mathcal{D}$-modules?
Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...
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