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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 …

Mathematicians sometimes use heuristics to form expectations about what might be true or false. For examples, see Matthew Emerton's answer to Why should I believe the Mordell Conjecture?, this blog po …

What are some good examples of algebraic theorems that have no known algebraic proofs? A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the th …

Historically, the boundary between pure mathematics and its applications was much less defined. However, with the increasing complexity of modern mathematics and the resulting need for specialization, …

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 …

What are instances of implicit reliance on countable or dependent choice in classic books? Two examples are Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald where it is claimed …

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 …

In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach …

I just saw a poster at the next office's door announcing the 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics. Here is another example of a meeting I will or …

I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some go …

This is a highly subjective question, but here goes. Has anyone ever published a result that was "taken seriously" by the research community, but was then discovered to be incorrect because of a "dumb …

What are some good references for GR from a categorical point of view? This is essentially just a big-list reference request. I'm aware that the subject exists and can do some basic sleuthing to fin …

Since the initial question was closed, but seems to be attracting a lot of discussion even after the fact, some of the comments qualifying as full answers IMO, I believe it is a reasonable question to …

Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\mathb …

I am currently doing a PhD in differential algebra and affine algebraic geometry at the University of Buenos Aires. I've been struggling to find a list of interesting and big open problems in these ar …