Unanswered Questions
49,215 questions with no upvoted or accepted answers
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Do we expect abelian varieties (and “Artin motives”) to generate the Grothendieck ring of varieties over a finite field?
The Tate conjecture implies that the category of motives over a finite field is generated (as tensor category) by the motives of abelian varieties and Artin motives. See [1] for details.
Let $K(\...
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768
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Wherefore art thou a Borcherds Product?
This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
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What to expect from spectral algebraic geometry
So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
- 3,019
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530
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Is this an $E_\infty$-algebra?
I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
- 40.3k
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Topological description of a blow up of a manifold along a submanifold
There's a very nice topological description of blow ups of complex manifolds at a point as connected sum with projective space. The following is an attmept to understand whether there's a higher ...
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370
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Cohomology with compact support for determinant varieties
I wonder if anyone knows anything about the cohomology with compact supports for determinantal varieties, such as the varieties of $m \times n$ matrices of full rank.
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547
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Relation between the Galois group of a polynomial and the symmetry of its roots
Let $f(x)$ be a polynomial with integer coefficients and non-zero discriminant (so each root of $f$ has multiplicity one). Consider the projective roots $\theta_1, \cdots, \theta_n$ of $f$, as ...
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What is an example of a non-abelian gerbe with connection?
Abelian gerbes can arise from obstructions to lifting a principal $C$-bundle to a principal $B$-bundle given some central extension $0\to A \to B \to C \to 0$ or as a representative of a cohomology ...
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Inter-Universal Teichmuller Theory and the Field with One Element
The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
- 3,309
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510
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On uniform Kazhdan's property (T)
For a finitely generated group $\Gamma$ and its finite generating subset $S$, the Kazhdan constant $\kappa(\Gamma,S)$ is defined to be
$$\kappa(\Gamma,S)=\inf_{\pi,v} \max_{g\in S} \| v - \pi_g v \|,$...
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Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?
Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
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602
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Precise form of the mean motion theorem
Consider an exponential polynomial
$$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\...
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Diffeomorphisms of $\mathbf R^n$
Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.
Question: Is there an example to ...
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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 $\omega$ ...
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Spaces locally modelled on $L^2(\mathbb R)$
In this recent question, I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (...
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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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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
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446
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The rank of a "triangle-free" matrix
This is a version of the question I asked recently, but the assumptions got now strengthened substantially.
Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
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Quasiperiodic continued fractions
Is anything known about continued fractions in which the sequence of integers is quasiperiodic?
Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
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599
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Open conjectures on the Fukaya category coming from physics
This is a slightly vague question (for which I apologize in advance): can somebody give examples to open conjectures on the behavior of the $Fuk(M,\omega)$ that come from string theory and can be ...
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Quantitative form of Wielandt's theorem
The following theorem was proved in [Helmut Wielandt. Eine Verallgemeinerung der invarianten Untergruppen. Mathematische Zeitschrift 45 (1939): 209-244.] a long time ago:
Theorem: (Wielandt) There ...
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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 ...
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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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511
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Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
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587
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Constructing a topos from a Heyting algebra
It is true that, given any topos $\mathcal{C}$ with a terminal object $1_\mathcal{C}$, $Sub(1_\mathcal{C})$ is a Heyting algebra.
Now suppose that we start with a Heyting algebra $H$. Is it always ...
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536
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Reference for equivariant Atiyah-Jänich theorem
The equivariant Atiyah-Jänich theorem is an isomorphism
$$
[X,F]_G \cong K_G^0(X),
$$
where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
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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 ...
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reference - Grothendieck on Thurston's work
In his 'dernières' years Grothendieck gets "interested" in Thurston's work.
"[...] je me suis intéressé ces dernières années - la géométrie hyperbolique à la Thurston et ses relations au groupe de ...
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517
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Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
- 16.2k
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365
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Admissible relations in a Banach algebra
Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...
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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 ...
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723
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Solving polynomial systems with homotopy. Where is the bottleneck?
I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions.
I want to solve this system numerically, but if I plug it ...
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Squeezing physics out of formal deformation quantizations
I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
- 5,407
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426
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Complexity classes for BSS machines
Given a first-order structure $\mathcal{S}$, a Blum-Shub-Smale machine on $\mathcal{S}$ is essentially a Turing machine where
Cells on the tape can hold arbitrary elements of $\mathcal{S}$.
The ...
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349
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Is there support for the term "Gelfand algebra"?
In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law
($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be ...
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Surprising approximate identity
While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik
found the following surprising approximate identity:
$$\ln{8\pi}\approx \pi\left[ 2\...
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430
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Counting abelian varieties over finite fields in a given isogeny class
Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
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475
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Maximizing the number of semistandard Young tableaux
Is anything known about the following question? Given a positive
integer $p$ and a real number $0<\alpha<1$, what partition $\lambda$
whose parts sum to $\alpha p^2$ (asymptotically) and whose ...
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Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
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Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
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On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$
(Major revision to incorporate new results in this MO cubic version.)
Note: All coefficients are in the rationals.
I. Cubic
In the linked post, it was shown that given a general cubic (via its ...
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What's the dimension of the space of CM cusp forms?
I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling.
Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
- 40.3k
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On some special spanning trees of grid graphs
I would like to know if there are existing results on the following objects:
spanning trees of a grid graph, with no corridor
where a corridor is a vertex having exactly two neighbors, on opposite ...
- 3,597
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877
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Chiral categories versus braided monoidal categories
Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...
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Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
- 648
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592
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What is the determinant of Poincaré duality?
For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant
$$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$
functorial with respect to quasi-...
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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 ...
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466
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"topological" Ochanine genus?
The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus, I am aware of a lift to a "spin orientation of Tate K-...
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669
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Is it possible to prove Mordell's conjecture geometrically?
Let $X/k$ be a smooth curve of genus $g>1$ over a number field $k$.
By the Faltings theorem (nee Mordell's conjecture), the set of $k$ - rational points
$X(k)$ is finite. Due to the Mordell-Weil ...
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648
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Countably decomposable von Neumann algebras
A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
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