All Questions
46,751
questions with no upvoted or accepted answers
35
votes
0
answers
905
views
Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
35
votes
0
answers
1k
views
Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
35
votes
0
answers
1k
views
Grothendieck's "List of classes of structures"
In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
34
votes
0
answers
692
views
Metrics on the 3-sphere with knotted geodesics
According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic (...
33
votes
0
answers
2k
views
History of the Proj construction in algebraic geometry
Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...
33
votes
0
answers
2k
views
Do there exist exotic 4-tori?
More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...
33
votes
0
answers
2k
views
Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
33
votes
0
answers
2k
views
Defining $\mathbb{Z}$ in $\mathbb{Q}$
It was proved by Poonen that $\mathbb{Z}$ is definable in the structure $(\mathbb{Q}, +, \cdot, 0, 1)$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by ...
33
votes
0
answers
2k
views
Peano Arithmetic and the Field of Rationals
In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is first order definable in $(\Bbb{Q}, +,...
33
votes
0
answers
1k
views
Subalgebras of von Neumann algebras
In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
32
votes
0
answers
1k
views
Cubic function $\mathbb{Z}^2 \to \mathbb{Z}$ cannot be injective
It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...
32
votes
0
answers
2k
views
The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
32
votes
0
answers
624
views
Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?
For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
32
votes
0
answers
3k
views
Vertex coloring inherited from perfect matchings (motivated by quantum physics)
Added (19.01.2021): Dustin Mixon wrote a blog post about the question where he reformulated and generalized the question.
Added (25.12.2020): I made a youtube video to explain the question in detail.
...
32
votes
0
answers
1k
views
Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?
In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.
QUESTION Is there any (added: ...
32
votes
0
answers
906
views
Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces
Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...
32
votes
0
answers
1k
views
Minimal number of intersections in a convex $n$-gon?
For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...
32
votes
0
answers
2k
views
Next steps on formal proof of classification of finite simple groups
While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson odd-...
32
votes
0
answers
2k
views
Is there software to compute the cohomology of an affine variety?
I have some affine varieties whose cohomology (topological, with $\mathbb{C}$ coefficients) I would like to know. They are very nice, they are all of the form $\mathbb{A}^n \setminus \{ f=0 \}$ for ...
32
votes
0
answers
958
views
Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
32
votes
0
answers
2k
views
Microlocal geometry - A theorem of Verdier
(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...
32
votes
0
answers
2k
views
A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"
Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$,
...
32
votes
0
answers
2k
views
$f\circ f=g$ revisited
This may be related to solving $f(f(x))=g(x)$. Let
$C(\mathbb{R})$ be the linear space of all continuous functions from
$\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
31
votes
0
answers
834
views
The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
31
votes
0
answers
2k
views
A question related to the Hofstadter–Conway \$10000 sequence
The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
31
votes
0
answers
893
views
Is this representation of Go (game) irreducible?
This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
31
votes
0
answers
1k
views
"Three great cocycles" in Complex Analysis as cohomology generators
In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...
31
votes
0
answers
1k
views
When are two C*-algebras isomorphic as Banach spaces?
We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
31
votes
2
answers
2k
views
Tiling of the plane with manholes
Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...
30
votes
0
answers
790
views
Interpretation of "1089-number trick" in terms of symmetric group action on cohomology group?
I tried posting the following on math.stackexchange, but no answers. I can of course delete if inappropriate.
The "1089 number trick" (see e.g. here) says that if you take a three-digit ...
30
votes
0
answers
869
views
Three real polynomials
Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
30
votes
0
answers
667
views
Do two integral matrices generate a free group?
Is it decidable whether two given elements of ${\rm GL}(n,{\mathbb Z})$ generate a free group of rank 2?
This is a simple question that I have been asking people for the past couple of years, but ...
30
votes
0
answers
893
views
On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
30
votes
0
answers
730
views
Cohomology of symmetric groups and the integers mod 12
When $n \ge 4$, the third homology group $H_3(S_n,\mathbb{Z})$ of the symmetric group $S_n$ contains $\mathbb{Z}_{12}$ as a summand. Using the universal coefficient theorem we get $\mathbb{Z}_{12}$ ...
30
votes
0
answers
989
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
30
votes
0
answers
3k
views
Greatly expanded new edition of a Bourbaki chapter on algebra?
Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
30
votes
0
answers
736
views
Is there an Ehrhart polynomial for Gaussian integers
Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
30
votes
0
answers
2k
views
Why do Clifford algebras determine $KO$ (and $K$-)-theory?
In the paper "Clifford modules" by Atiyah-Bott-Shapiro, they construct a family of Clifford algebras $C_k$ over the real numbers, so that $C_k$ is the algebra associated to a negative definite form on ...
30
votes
0
answers
2k
views
What do dessins tell us about the absolute Galois group?
I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single ...
30
votes
1
answer
2k
views
When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
29
votes
0
answers
891
views
Todd class as an Euler class
Let $X$ be a relatively nice scheme or topological space.
In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. ...
29
votes
0
answers
2k
views
Did Grothendieck overestimate topoi?
I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines:
Из этих тем ...
29
votes
0
answers
1k
views
A modern perspective on the relationship between Drinfeld modules and shtukas
Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ...
29
votes
0
answers
1k
views
Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?
Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
29
votes
0
answers
1k
views
Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
29
votes
0
answers
977
views
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 ...
29
votes
0
answers
3k
views
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 ...
29
votes
0
answers
3k
views
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
739
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 ...