Unanswered Questions
49,207 questions with no upvoted or accepted answers
16
votes
0
answers
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
16
votes
0
answers
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 ...
- 13.4k
16
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0
answers
330
views
Is tightness decidable?
Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...
- 18.7k
16
votes
0
answers
1k
views
How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?
Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$
The functor $E^*_{S^1}(-)$ takes in a space $X$ ...
- 3,539
16
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0
answers
854
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Self-avoiding random walks that always turn
I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...
- 151k
16
votes
0
answers
591
views
Lifting DG-categories to characteristic zero
The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
- 25.6k
16
votes
0
answers
399
views
Power series which are $p$-adic modular forms for all $p$
Suppose that, for some integer $k$, a series $f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$ has the property that for every prime $p$, $f(q)$ is the $q$-expansion of a $p$-adic modular form of weight $k$ ...
- 3,910
16
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0
answers
542
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$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
16
votes
0
answers
423
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Retractions of Yoneda are reflectors, i.e., left adjoints?
Background
It is fairly well known that if a full subcategory $i: C \hookrightarrow D$ has a left adjoint $F: D \to C$, then the canonical counit $F i(c) \to c$ is an isomorphism. (A classical ...
- 53.3k
16
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answers
1k
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"extended TQFT" versus "TQFT with defects"
There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...
- 1,219
16
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0
answers
809
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Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
16
votes
0
answers
444
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Does $S^4$ have a "symplecto-homeomorphic" structure?
The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_\text{std})\to(\mathbb{R}^4,\omega_\text{std})...
- 17.5k
16
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0
answers
650
views
Approximating homeomorphisms of 2-disk by diffeomorphisms
Any homeomorphism of a compact surface can be approximated by diffeomorphisms.
Is there a parametrized version of this result, where the parameter space is an $n$-disk?
In other words, if $S$ is a ...
- 29.1k
16
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0
answers
448
views
Are dualizable modules finitely generated?
Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...
- 161
16
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0
answers
664
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real algebraic geometry software?
Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
- 96.4k
16
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0
answers
2k
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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
- 13.5k
16
votes
0
answers
878
views
What is the maximal Picard number of a surface in P^3?
Let $X$ be a nonsingular surface in $P^3$ of degree $d$. Then the Picard number does not exceed $h^{1,1}\approx 2/3 d^3$, whereas the maximal number attained that I know is that of a Fermat surface, ...
- 4,959
16
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0
answers
1k
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How many sporadic simple groups are there, really?
I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...
- 6,290
16
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0
answers
298
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Realization spaces of 3-dimensional polytopes with fixed face areas
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
- 31.2k
16
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0
answers
454
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A Product Related to Unrestricted Partitions
Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
- 2,143
16
votes
0
answers
879
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L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
- 643
16
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0
answers
539
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Identification of a curious function
The following question was asked on MSE but there were no replies.
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\...
- 1,906
16
votes
0
answers
784
views
How to explain the picturesque patterns in François Brunault's matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...
- 19.6k
16
votes
0
answers
2k
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Grothendieck 's question - any update?
This question is migrated from math.stackexchange. I ask because it is still unclear to me and I did not receive an answer.
I was reading Barry Mazur's biography and come across this part:
...
- 6,259
16
votes
0
answers
596
views
What are some examples of weak ω-categories?
As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...
- 5,309
16
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0
answers
861
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Is "being a full ring of quotients" a Morita invariant property?
Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
- 2,454
16
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0
answers
373
views
What is the simplest known arithmetic definition of exponentiation?
For $n$ a natural number, let $E_n$ denote the set of bounded arithmetic formulas consisting of n alternating blocks of bounded quantifiers starting with an existential quantifier, followed by a ...
- 521
16
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0
answers
546
views
What can be the dimension of a pointless smooth proper Z-scheme?
What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?
Of course, there ...
- 149k
16
votes
0
answers
671
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Does the Ricci flow approach shed light on Poincare conjecture in higher dimension?
I know that the Poincaré conjecture was first proved in dimension ≥ 5, then dimension 4, and finally 3. I'm just curious, does the Ricci flow approach by Perelman shed any light on the high dimension ...
- 3,806
16
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1k
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Homology classes of subvarieties of toric varieties
Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety.
Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero?
Background
If $X$ is a Kaehler variety, this is of ...
- 16.2k
16
votes
0
answers
1k
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What is the appropriate setting for Cauchy's Integral Formula?
For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = \...
- 1,124
16
votes
0
answers
645
views
Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups
In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...
- 3,351
16
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0
answers
1k
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Structure of the algebra of mod $p$ modular forms
Let me first define the algebra $M$ I am talking about: let us fix a prime $p$, an integer $N$
not divisible by $p$. For $k$ an integer, let me call $N_k$ the $\mathbb{Z}$-module of modular
forms of ...
- 26k
16
votes
0
answers
824
views
Capelli determinant = Duflo ( determinant) - was it known ?
Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$).
I googled a lot ...
- 24.9k
16
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0
answers
2k
views
When does a correlated Brownian motion leave a square?
Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some
constant $\rho \in [-...
- 16.5k
16
votes
0
answers
558
views
Catalan objects associated to a univariate polynomial
Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ ...
- 6,292
16
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0
answers
1k
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Regularizing the divergent sum $1^k + 2^k + \cdots$
EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...
user11000
16
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0
answers
11k
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Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
16
votes
0
answers
851
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Applications of Berger's Curvature Estimate
I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor:
Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the ...
- 4,899
16
votes
0
answers
4k
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Sheaf cohomology and inverse limits
In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13:
Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
- 25.6k
16
votes
0
answers
2k
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Characterizing the surcomplex numbers
Conway showed that the Field of surreal numbers ("${\bf No}$")
is the maximal totally ordered Field.
Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is
the universally ...
16
votes
0
answers
763
views
Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces
Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.
Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
- 31.5k
16
votes
0
answers
530
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Comparing the Kazhdan-Lusztig and Steinberg pre-orders
Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
- 44.7k
16
votes
0
answers
2k
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The radius of convergence of the p-adic exponential function.
As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is
$$\rho = p^{-1/(p-1)}.$$
This is typically proven by computing ...
- 13.3k
16
votes
0
answers
668
views
Dual of the Ultraproduct of a Banach Space
Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like":
$(E_i^*)_U$, the ultraproduct of the duals of the ground spaces.
The space made up ...
- 7,145
16
votes
0
answers
605
views
Division fields of abelian varieties over function fields
Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
- 1,673
16
votes
0
answers
2k
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Can one compare integral structures on de Rham and crystalline cohomology?
Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology,
$H^i_{\...
- 37k
16
votes
0
answers
1k
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representation theoretic interpretation of Jack polynomials
Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization ...
- 4,466
16
votes
0
answers
910
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Polynomials with presumably positive coefficients
After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...
- 13.5k
16
votes
0
answers
1k
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Galois theory timeline (II)
This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
- 12.6k