MathOverflow Weekly Newsletter
MathOverflow Weekly Newsletter

Top new questions this week:

What is the purpose of the flat/fppf/fpqc topologies?

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully. As someone who originally started in ...

ag.algebraic-geometry grothendieck-topology  
asked by Simon Rose 22 votes
answered by Tyler Lawson 28 votes

Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$. Question: Let $n$ be a ...

nt.number-theory diophantine-equations  
asked by Stefan Kohl 20 votes
answered by Leonardo 13 votes

Categorical proof subgroups of free groups are free?

This is a crossport of this question from MSE. Is there a categorical proof that subgroups of free groups are free? How about the result that subgroups of free abelian groups are free abelian? ...

gr.group-theory ct.category-theory free-groups abelian-groups  
asked by Exterior 18 votes
answered by Qiaochu Yuan 12 votes

A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, ...

nt.number-theory co.combinatorics  
asked by Clark Kimberling 15 votes
answered by Michael Stoll 10 votes

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...

ag.algebraic-geometry co.combinatorics graph-theory mg.metric-geometry  
asked by Adam P. Goucher 15 votes
answered by Ilya Bogdanov 10 votes

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...

rt.representation-theory noncommutative-algebra invariant-theory determinants  
asked by semyon alesker 14 votes
answered by Zurab Silagadze 4 votes

Why is "The Higman Rope Trick" thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma: If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...

gr.group-theory terminology combinatorial-group-theor  
asked by ShlomiF 13 votes
answered by Flounderer 2 votes

Greatest hits from previous weeks:

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

books ca.analysis-and-odes big-list textbook-recommendation real-analysis  
asked by Ryan 10 votes
answered by lhf 13 votes

Computer Algebra Errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers. After ...

computer-algebra big-list  
asked by Kevin O'Bryant 67 votes
answered by Dan Piponi 120 votes

Can you answer these?

Exactness of pure functors

I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman": Lemma. A pure functor is exact. Definitions: A mixed category $\mathcal{M}$ is a ...

motivic-cohomology abelian-categories  
asked by Mostafa 6 votes

Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic

Let $X$ be a complex space. We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$. We say that $X$ is Kobayashi hyperbolic if the Kobayashi ...

complex-geometry kobayashi-hyperbolicity  
asked by diverietti 4 votes

Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup

Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety. Let ...

ag.algebraic-geometry algebraic-groups homogeneous-spaces  
asked by Mikhail Borovoi 4 votes
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