**4**

votes

**1**answer

183 views

### How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic

Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.
If $k= \mathbb C$, there is a natural injective morphism of vector spaces
...

**7**

votes

**0**answers

175 views

### Mapping class group action on fundamental group of punctured elliptic curves

Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.
...

**13**

votes

**2**answers

373 views

### What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...

**3**

votes

**1**answer

154 views

### Jaffe's exact sequence

Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow ...

**1**

vote

**1**answer

36 views

### Spacing of the largest singular values of Wishart matrix

Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...

**6**

votes

**0**answers

177 views

### How do the direct and inverse image sheaf functors interact with homotopy?

This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by ...

**1**

vote

**0**answers

22 views

### What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes. For an infinitely divisible ...

**10**

votes

**3**answers

928 views

### is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space.
Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry.
We know ...

**0**

votes

**0**answers

21 views

### Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...

**1**

vote

**0**answers

19 views

### Relation between indexed languages (OI-macro or context-free tree) and scattered context languages

I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by
scattered context grammars--J Hopcroft).
I think that ...

**195**

votes

**7**answers

97k views

### Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

**-1**

votes

**0**answers

20 views

### Strong solution and measurability on Ikeda and Watanabe context

This question is from Chap 4 of Ikeda and Watanabe - Stochastic differential equations and Diffusion processes pg 149
and on page 152
I believe that in the context of the last paragraph ...

**143**

votes

**16**answers

57k views

### What's a mathematician to do? [closed]

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ...

**0**

votes

**0**answers

16 views

### Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation [on hold]

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...

**6**

votes

**0**answers

83 views

### Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for ...

**7**

votes

**4**answers

681 views

### Is the Manickam-Miklós-Singhi Conjecture solved?

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...

**2**

votes

**1**answer

82 views

### Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.
I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...

**1**

vote

**0**answers

19 views

### Solution to Helmholtz equation with non-circular boundary

Let $D$ be an homogeneous 2D domain with non-circular boundary $\partial D$.
I am trying to solve the Helmholtz equation
$$
\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)
$$
in which ...

**8**

votes

**1**answer

489 views

### Geometric meaning of the black hole horizon

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...

**39**

votes

**15**answers

5k views

### What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...

**37**

votes

**12**answers

15k views

### Theorem versus Proposition

As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main ...

**0**

votes

**0**answers

113 views

### Changes in singular Values of matrix when adding row

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is:
Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...

**2**

votes

**1**answer

149 views

### Two questions on substitutability

(1) The condition that a term $a$ be substitutable for another term in an expression can be given a recursive definition. Who first developed such a definition?
(2) One sometimes see the phrase "$a$ ...

**44**

votes

**8**answers

3k views

### Has anyone thought about creating a formal proof wiki with verifier?

Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc. I imagine there must be many further ...

**0**

votes

**0**answers

13 views

### The general roles of a Kernel and multiplier in transformations [on hold]

What are the general roles of a kernel and multiplier in functional transformations, such as the Laplace transform? I am asking this question because I have seen these terms used in more than one ...

**55**

votes

**4**answers

7k views

### Etale cohomology — Why study it?

I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...

**1**

vote

**1**answer

211 views

### Proof of regularity for bounded elliptic problem

We consider the boundary value problem for potential in the form:
$$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$
with boundary conditions
$$\nabla ...

**30**

votes

**2**answers

3k views

### Intuition for coends

Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end
$$
\int^{c\in C} F(c,c)
$$
as the co-equalizer of
$$
\coprod_{c\to ...

**44**

votes

**1**answer

3k views

### Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...

**8**

votes

**5**answers

3k views

### Has anyone found an error in an early version of Neukirch?

I remember a friend in graduate school throwing an early edition of Jurgen Neukirch's Algebraic Number Theory book against a wall (so hard that it split the binding) after he had worked for a number ...

**13**

votes

**2**answers

755 views

### What do correlation functions compute in CFT?

I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power ...

**6**

votes

**2**answers

919 views

### Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...

**34**

votes

**1**answer

11k views

### What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...

**8**

votes

**0**answers

176 views

### Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the ...

**12**

votes

**3**answers

1k views

### How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstrass form?

Consider the Weierstrass cubic
$$y^2z = x^3 + A\, xz^2+B\,z^3.$$
This defines a curve $E$ in $\mathbb{P}^2$, which if smooth is an elliptic curve with basepoint at $[0,1,0]$.
I'm interested in having ...

**2**

votes

**2**answers

620 views

### All properties of a mathematical object

This is primarily a question about related literature. I am looking for specific references, or terminology that I can use to search for references.
Let A a well defined mathematical structure of ...

**15**

votes

**1**answer

637 views

### Topos associated to a category

For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally ...

**4**

votes

**4**answers

216 views

### Bound on the number of unlabeled cographs on n vertices

A cograph is a graph without induced $P_4$ subgraphs. I am looking for a reference for a simple exponential bound on the number of distinct unlabeled cographs on $n$ vertices. By the Mathworld ...

**1**

vote

**1**answer

525 views

### Proof Sketch: The pullback of the inclusion of the 0th vertex into the standard n-simplex by a right fibration is a deformation retract (450 point bounty if answered by 2am EST)

I was not sufficiently clear on my last attempt at asking a similar (but not identical) question. Tom Goodwillie mentioned (in the accepted answer) that the question can be reduced to this one and ...

**6**

votes

**1**answer

182 views

### vector bundles associated to a covering space

Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...

**1**

vote

**0**answers

15 views

### Constrained absolute orientation of 3D point sets

Let us assume we have two 3D point sets, $P=\{p_i\}$ and $Q=\{q_i\}$, and that we need to recover the transformation that takes $P$ as close to $Q$ as possible. In particular, I am interested in ...

**0**

votes

**0**answers

36 views

### Is the localization sequence exact in the middle mod. algebraic equivalence?

Let $X$ be a smooth projective $k$-variety ($k=\bar k$) and $U\subset X$ an nonempty open subset. Is it true that a cycle algebraically equivalent to zero in $U$ comes from a cycle of $X\backslash U$ ...

**32**

votes

**12**answers

3k views

### Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...

**3**

votes

**0**answers

78 views

### Langlands-Shahidi method carried out in the simplest case?

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function. Could the constant term of GL(2)-Eisenstein series to prove any ...

**1**

vote

**0**answers

46 views

### Where does the algebraic closure enter into Block's Theorem?

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in ...

**2**

votes

**0**answers

44 views

### space of stability condition for an elliptic curve

Let $E$ be an elliptic curve. I want to understand why $\mathrm{Stab}(E)/\mathrm{Aut}(D^b(E))$ is a $\mathbb{C}^\times$-bundle over the moduli space $\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ of elliptic ...

**-4**

votes

**0**answers

27 views

### Elementary Expected Value Question [on hold]

Let $N$ be a positive integer. A soon to be bankrupt
casino lets you play the game $G(N)$. In the game $G(N)$, you roll a
typical, fair, six-sided die, with faces labeled 1 through 6, up to $N$
times ...

**4**

votes

**2**answers

203 views

### What is the fastest algorithm for counting points in elliptic curves mod n?

I need an algorithm for getting the order of the group in random elliptic curves mod n, being n a composite module. As far as I know, usual algorithms like Schoof's algorithm only works with prime ...

**3**

votes

**1**answer

51 views

### Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu_n$, the distributions of ...

**7**

votes

**0**answers

76 views

### Reduction formula for Schubert polynomials

In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning ...