Open-problem statements, with their sources attached.

The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.

Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$, $|S| \gg K^{-O(1)} |A|$, with $S^8 \subset A^4$?

Upper bound for $c(p)$ for $1 < p \le \infty$, improving the best-known value at $p = \infty$.

Let A ⊂ R be a set of positive measure. Does $A$ contain an affine copy of {1, 1/2, 1/4, . . . }?

Erdős Problem 598:* Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X \subseteq m$ with $|X| = \kappa$ contains subsets of all possible colours?

Is it true that for all entire functions f = ∑ aₖzⁿₖ such that ∑' 1 / nₖ < ∞, limsup (fun r => ratio r f) atTop = 1?

Is $e + \pi$ irrational?

(C) Breakdown of Navier–Stokes solutions on ℝ³.

A stronger version of the MLC conjecture, stating that all multibrots are locally connected. Note that we don't need to require 2 ≤ n because the conjecture holds in the trivial cases n = 0 and n = 1 too.

The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\infty$ converges.

Problem 4.2.* Let $\Omega \subset \mathbb{R}$ be a finite union of three or more intervals. If $\Omega$ weakly tiles its complement, must it also tile its complement properly?

Conjecture 1.6 (Odd case).* For a nonempty isolate-free graph $G$ on $n$ vertices, if $D$ is odd, then $(D + 1)(D + 3) \cdot i(G) \leq (D^2 + 3) \cdot n$.

What is the size of the largest $A \subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x$, $y$, $z$ from $A$, at least two of the distances $|x - y|$, $|y - z|$, $|x - z|$ are equal.

Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.

Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?

Strong Sensitivity Conjecture, for every Boolean function f : {0,1}^n → {0,1}, bs(f) ≤ s(f)^2.

We call this the strong sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial bound in terms of s(f). Huang's celebrated result (often called the sensitivity theorem) gives a quartic bound, bs(f) ≤ s(f)^4, thereby settling the original conjecture.

Conjecture 4.1*: For any prime $k$ larger than $5$, $S(k) > k$.

Probably there is no such $A$ for the polynomial $X^k$ for any $k \ge 2$. This is asked in [Sek59].

Benchmark open subproblem: existence of a SIC-POVM in dimension $60$.

The Köthe conjecture: every left nil radical is contained in the Köthe radical.