# All Questions

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### 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 ...
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### 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 $\widehat{SL_2(\mathbb{Z})}$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 "$a$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $\neg CH$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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### 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 ...
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### 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 $X\backslash U$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...