Open-problem statements, with their sources attached.

Does there exist a category that is pretriangulated but not triangulated?

There exists a proper ideal I in a (commutative) total ring R of fractions that is an invertible module. If I ⊊ R is such an example, I must have infinite order in the Picard group, and R must not be Noetherian (otherwise it must be semi-local and therefore have trivial Picard group).

Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts relating it to EuclideanSpace ℂ (Fin 3)?

Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ hold when both suprema are finite?

Is there any polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that $f : \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ is a bijection?

Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients. If $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ are also $0, 1$ polynomials.

Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

Conjecture 7 from Kahn–Kalai 2006: the same statement as the original conjecture, but with the additional assumption that $t$ is the critical probability for $F$, namely $\mu_t(F) = 1/2$.

If $2^x$ and $3^x$ are integers, then $x$ must be an integer.

Is 2n the complexity of 2^n for 0 < n?