Open-problem statements, with their sources attached.

There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$. This is Conjecture 6.3 in [Har1974].

Problem 4.1.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\mathrm{supp}(\nu)$ have bounded density?

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?

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.

Problem 4.3.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\nu$ be expressible as a convex combination of proper tiling measures?

Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?

De Giorgi's conjecture holds in dimension $n ≤ 8$.

Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact hausdorff space as an image of a homogeneous compact space under a continuous mapping?

Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators with shape parameter $\alpha > 0$, $\alpha \neq 1$.

The source asks for sufficiently smooth functions. This concrete version uses ContDiffOn ℝ 2 f I as a readable baseline regularity assumption; since the domain is the compact interval $[0,1]$, this also explains why no separate boundedness assumption is included here. The variants below record the unknown smoothness threshold more explicitly.

De Giorgi's conjecture holds in dimension $n = 4$.

Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact hausdorff space first countable?

Markel's $S_3$-conjecture* (1973): any nontrivial finite ah-group is isomorphic to $S_3$.

The conjecture is open in general; it is known to be true for solvable groups.

Variant of the Bézier-Bernstein Voronovskaja problem which treats "sufficiently smooth" as an eventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every $C^m$ function on $[0,1]$ should have the asserted asymptotic formula.

De Giorgi's conjecture holds in dimension $n = 5$.

Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠?

Existence-only version of the eventual-smoothness variant. This separates the first part of the source problem, proving that the scaled sequence has some limit, from the stronger task of finding an explicit expression for that limit.

De Giorgi's conjecture holds in dimension $n = 6$.

Problem 17 in [Ar2013]: Is it true that every nonempty ω-monolithic compact hausdorff space contains a point with a first countable neighborhood basis?

Note: Nonempty X is required since the conclusion asserts the existence of a point.

The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.