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Here's an example of a pair of completely unrelated (as far as I can tell) sequences: Maximum order of an explicit Runge–Kutta method with n function evaluations in each step (A187103) and the metric …

Here is a version of the table of contents of Bowditch's lecture notes A course on geometric group theory. Group presentations, free groups, abelianisation. Cayley graphs. Quasi-isometries and thei …

Only for $m=2$, $\sum_{i=1}^mx^{2n+1}_k$ is always divisible by $\sum_{i=1}^mx_k$. Indeed, $$x+y | x^{2n+1}+y^{2n+1}, n \in \mathbb N.$$

I don't have 10 things for your list, but I can describe the syllabus of the Introduction to Geometric Group Theory Masters' course I have taught for the last couple of years. I hope this is worth men …

This question is seven years old, but since I wanted to ask the same question and provide those examples for induction on prime numbers, I found this question and saw that the first example is missing …

Here is my take. Unlike Andy, I would not structure such a course around big theorems. In part, this is because your students simply do not have enough background to handle any "big theorems." Instead …

Geometric group theory is a huge subject, and a course that really tried to cover all of it would be too disjointed to be useful. If I were teaching such a course, I would choose a few major theorems …

What are the ten most fundamental topics in geometric group theory? This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expecte …

A trick that is used daily is Zorn's Lemma. Sure, every mathematician knows it, but it certainly helped prove non-trivial propositions and it is in daily use. I would consider it in a list of the Top …

The trivial cohomology box trick When trying to solve a problem, prove that: -- the obstruction to the existence of a solution lives in a cohomology box (a cohomology space, or group, or set), - …

A trick/technique that I like (and used) a lot is the formal geometry approach (after Gelfand-Kazhdan) for passing from a local to a global result. Let $X$ be a $d$-dimensional manifold. There is a an …

In my personal field, applied optimal transport for PDEs, we often play the following game, so much so that some of my colleagues and I actually call it Brenier's trick (after Yann Brenier): When mini …

4 is the smallest composite number.

Here is the abstract of Andreas Reinhart, A counterexample to the conjecture of Ankeny, Artin and Chowla, available at https://arxiv.org/html/2410.21864v1 ``Let $p$ be a prime number with $p\equiv1\bm …

In all 200+ pages of my category theory notes, there were essentially three tricks I used in proofs: Prove that two arrows are both the arrow induced by a universal property, so they're the same arro …