No starter "accessible" well known open problems

Question

We all know very famous open problems. Usually the ones that become famous are well studied and lots of progress was achieved and conjectures, partial results, reductions and so on exist. This is one of the reasons they become famous. This kind of problems are usually hopeless to attack without lots of investment of time and dedication to learn the basics and beyond.

I am interested in the other extreme, that is, I would like relatively easy to state problems which are well known (at least in the relevant area), but no one has a clue how to start solving them, preferably no one even has a good guess what the right answer is.

Let me give you two examples from my research area:

1. A pro-$p$ group in which there exists a bound on the number of generators (topologically) for all closed subgroups is said to have finite rank. These are well known due to the works of Lazard, and Lubtozky and Mann, for instance, they are $p$-adic analytic. However, as far as I know known no-one has a clue whether a pro-$p$ group in which every closed subgroup is finitely generated has finite rank? In other words, if a pro-$p$ group is not of finite rank does it contain a closed subgroup which is not finitely generated?

2. Zelmanov solved the Restricted Burnside Problem that is there are finitely many finite $p$-groups with $d$-generators and exponent at most $e$. This is equivalent to saying that every finitely generated pro-$p$ of finite exponent is finite. He also generalized and proved the Burnside Problem for pro-$p$ groups, that is, every finitely generated torsion pro-$p$ is finite. However, the Generalized Burnside Problem for pro-$p$ group is completely open, that is, is it true that a torsion pro-$p$ group (not necessarily f.g.) has finite exponent? (I'll just comment that Burnside kind of problems are usually very hard.)

Motivation: Well I recently retired and while I do not have the energy (or talent) to try to solve most famous open problem, it would be nice to have a list of famous problems that either by luck one might encounter a counter example or by luck one might find a very different approach.

Answer 1

Since I think the question has a reasonable interpretation, let me get the ball rolling with an open problem where I am not aware of any partial progress or proposals for a proof/counterexample.

OPEN PROBLEM. Let $H^\infty$ denote the Banach space of all bounded analytic functions on the open unit disc, equipped with the supremum norm. Does $H^\infty$ have Grothendieck's approximation property?

Concretely: is it the case that for each norm-compact subset $K\subset H^\infty$ and each $\varepsilon>0$, we can find a finite-rank operator $T:H^\infty\to H^\infty$ such that $\sup_{x\in K} \lVert T(x)-x\rVert \leq\varepsilon$?

Answer 2

How about:

OPEN PROBLEM. Let $k$ be a field. Describe the group of automorphisms of the polynomial ring $k[x_1,\ldots,x_n]$, where $n \geq 3$.

This is a seemingly basic problem in algebra. The automorphism groups of polynomial rings in two or fewer variables are understood. For three or more variables, I don't think there is anything close to a conjecture of what they should look like.

Answer 3

OPEN PROBLEM. Is the classical permutation group $S_N$ a maximal quantum subgroup of the quantum permutation group $S_N^+$?

This is true for $N\leq 5$:

• $N\leq 3$: in this range $S_N^+=S_N$,
• $N=4$: Banica and Bichon [1], through classifying the quantum subgroups $\mathbb{G}\subseteq S_4^+$ , noted that $S_4\subseteq S_4^+$ is a maximal quantum subgroup, and conjectured that $S_N\subseteq S_N^+$ is a maximal quantum subgroup at all $N$.
• $N=5$: Only recently did Banica [2] use advances in subfactor theory to show that $S_5\subseteq S_5^+$ is also a maximal quantum subgroup.

My understanding is that the $N=4$ approach is intractable for $N>4$ and the $N=5$ approach should also be intractable for $N>5$.

Relative to the age of the field, this is a long-standing problem. There are advances on a restricted version of the conjecture: there are no so-called easy quantum group counterexamples. I have published some very (very) basic properties of any counterexample. There is little or no published failed attempts as far as I know.

More detail can be added if required.

[1] T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626, 74–114 (2009)

[2] T. Banica, Homogeneous quantum groups and their easiness level, Kyoto J. Math. 61, 1–30 (2021)