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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 …

I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to …

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of topo …

Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categorie …

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection of tricks u …

One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone seque …

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems I just noticed it while trying to understand his answer. But indeed it wo …

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where 'r …

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert have only a few tricks …

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be beauti …

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P fro …

Inspired by When is 2 qualitatively different from 3? Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being re …

Conjectures play important role in development of mathematics. Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch …

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 in …

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 woul …