# All Questions

**4**

votes

**1**answer

93 views

+50

### Ascertain properties of a new kind of rectilinear-convex set

PREABMLE TO MY QUESTION
I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many ...

**1**

vote

**0**answers

54 views

### Maximal $k$-split $k$-tori are $G(k)$-conjugate, but maximal $k$-tori are not?

Suppose $G$ is a reductive algebraic group defined over a field $k$.
Is this right?
If so, it means for each maximal $k$-split $k$-tori $T$ there are many maximal $k$-anisotropic ones $T'$ that ...

**6**

votes

**1**answer

157 views

+200

### Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...

**8**

votes

**0**answers

144 views

+50

### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question.
In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...

**3**

votes

**1**answer

96 views

### Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle

I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose ...

**7**

votes

**0**answers

60 views

### Near model completeness

A theory $T$ is called near model complete if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of ...

**7**

votes

**0**answers

119 views

### A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...

**5**

votes

**1**answer

149 views

+50

### Derive an orthonormal system by Riesz basis $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$

Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform
$$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$
where
...

**-4**

votes

**0**answers

33 views

### Simple Regular Matrix proof Problem [on hold]

So I got this question:
AB-I is Regular a regular matrix.
Provide a proof that BA-I is also a regular matrix.
Thanks

**2**

votes

**1**answer

74 views

### Curvature computations of globally symmetric spaces of rank $1$

I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T_pM ...

**5**

votes

**0**answers

79 views

### model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...

**60**

votes

**47**answers

9k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**3**

votes

**0**answers

154 views

+50

### Castelnuovo -Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be graded ring and $I,J$ be quartic square free monomial ideals in $R$. Let $I=\langle n_1,\ldots,n_s \rangle$ and $J=\langle m_1,\ldots ,m_t\rangle$
Definition:
...

**0**

votes

**0**answers

41 views

### Hashimoto Matrix (Non-backtracking operator) and the Graph Laplacian

The question is: how can we recover the graph Laplacian or its spectrum from the Hashimoto Matrix (also commonly called the Non-backtracking operator)?
To make the question as self-contained as ...

**7**

votes

**0**answers

53 views

### Embedding of parallelizable closed smooth manifold

It seems that any closed parallelizable smooth $n$-manifold can be immersed in the $(n+1)$-dimensional Euclidean space (compared to the general $(2n-1)$-dimensional case). Is there any analogous ...

**12**

votes

**2**answers

478 views

+50

### Braid group on 4 strands

Consider the braid group $B_4$ with generators
$a=\ $,
$b=\ $
and $c=\ $
Assume
$$\alpha, \alpha'\in \langle a,c\rangle\backslash(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle ...

**96**

votes

**17**answers

23k views

### Mathematical software wish list

Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has ...

**3**

votes

**2**answers

324 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**-4**

votes

**0**answers

104 views

### Mathematics textbooks with long title [on hold]

L'Hôpital's book has the long title
Analyse des infiniment petits pour l'intelligence des lignes courbes,
and Ya. Perelman's Algebra Can be Fun mentions an old Russian Mathematics textbook with the ...

**6**

votes

**0**answers

22 views

### Does non-stablity imply that there is a difference between non-forking and coheir extension

Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$.
Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.
Also ...

**20**

votes

**1**answer

347 views

### Why does McMahon formula look like the inclusion-exclusion principle?

The McMahon formula for the number of tilings of an $a \times b \times c$ hexagon by lozenges:
$$ \Big[H(a)H(b)H(c)\Big] \Big[H(a+b)H(b+c)H(c+a)\Big]^{-1} \Big[H(a+b+c)\Big]$$
looks oddly like the ...

**2**

votes

**0**answers

76 views

### Enumerating factors in intervals

Given $1<a<N-N^{1/\alpha}$ where $\alpha\geq2$, denote the number of distinct factors of $N$ in $[a,a+N^{1/\alpha}]$ as $\sigma_{0,a}(N,\alpha)$ denote ...

**8**

votes

**2**answers

336 views

### Is every finitely generated flat modules projective over a commutative ring with a finite number of minimal primes?

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective.
If we notice that for each minimal prime $p$ of the ring, ...

**0**

votes

**0**answers

145 views

### Hodge structure of the cohomology of a complement

What is the hodge structure given to the cohomology of the complement of a closed subset with respect to a smooth variety?
This is not quite what is wanted. In fact,
The hodge structure is ...

**12**

votes

**0**answers

239 views

### 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 ...

**6**

votes

**1**answer

149 views

### Behaviour of $\zeta(1-it)/\zeta(1+it)$?

I am trying to understand the behaviour of
$$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...

**3**

votes

**1**answer

77 views

### Mixed norm estimate for the heat equation

Consider the inhomogeneous linear heat equation
$$\partial_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...

**13**

votes

**2**answers

964 views

### Derived algebraic geometry: how to reach research level math?

I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.
My goal is to study derived algebraic geometry, where derived ...

**10**

votes

**0**answers

153 views

### 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[ ...

**36**

votes

**1**answer

562 views

### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...

**7**

votes

**3**answers

306 views

### Computations in modular cohomology of finite groups

Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...

**9**

votes

**1**answer

163 views

### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...

**6**

votes

**2**answers

422 views

### What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...

**192**

votes

**97**answers

32k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**0**

votes

**1**answer

145 views

### Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification.
Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...

**1**

vote

**0**answers

191 views

### Why do some people adamantly insist on 'toposes' instead of 'topoi'? [closed]

I've heard that several category and topos theorists, first and foremost Johnstone (see the comments to this question) adamantly insist on 'toposes' as the plural of 'topos'. I was wondering whether ...

**250**

votes

**15**answers

34k views

### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...

**2**

votes

**1**answer

34 views

### Bound for the generalised Rényi dimension of a measure

If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for ...

**4**

votes

**2**answers

165 views

### Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have
$$
\mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*,
$$
where $\lambda$'s are dominant weights. Let $U^-$ be the ...

**1**

vote

**0**answers

26 views

### Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence:
Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...

**5**

votes

**0**answers

134 views

### Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$.
Studying some branched covers of $A$, I was led to consider rank $2$ holomorphic vector ...

**10**

votes

**5**answers

508 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...

**2**

votes

**0**answers

74 views

### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

**231**

votes

**72**answers

89k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**4**

votes

**2**answers

170 views

### Reference for (co)limit-preserving functor $X\mapsto R^X$

Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor ...

**206**

votes

**67**answers

99k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**-4**

votes

**0**answers

80 views

### An idea of a matrix product I had in mind

There's a matrix product which I had in mind for sometime now, and I would like to know if someone has thought about it before, it's really simple, not sure for its applications.
Here it is for ...

**28**

votes

**0**answers

322 views

### Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...

**28**

votes

**2**answers

1k views

### When is there a submersion from a sphere into a sphere?

(First posted on math.SE, with no answers.)
That is:
For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
The discussion at this math.SE question has ...

**107**

votes

**96**answers

60k views

### Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...