Open-problem statements, with their sources attached.

The Banach--Mazur rotation problem asks whether every separable Banach space whose group of linear isometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert space.

Show that every bounded linear operator T : H → H on a separable Hilbert space H of dimension at least 2 has a non-trivial closed T-invariant subspace: a closed linear subspace W of H, which is different from H and from {0}, such that T ( W ) ⊂ W. One needs the assumption that the dimension of H is at least 2 because otherwise any subspace would be either H or {0}.

Gerver's sofa is the unique sofa that attains the sofa constant.

$F(n) \leq n^{3/2}$.

Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances?

Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?

The four dimensional case of the smooth version of the conjecture is still open. See [Wang2017].

Does every finite partially ordered set that is not totally ordered contain two elements $x$ and $y$ such that the probability that $x$ appears before $y$ in a random linear extension is between $\frac 1 3$ and $\frac 2 3$?

The set of all total order extensions is represented as order preserving bijections $P$ of $1, ..., n$.

Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?

P ≠ NP*:

The conjecture that the complexity classes P and NP are not equal.

The negation of Finite.Equation677_implies_Equation255.

Probably this is true. It would be a stronger form of Equation677_not_implies_Equation255.

Discussion thread here: https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255

If $n = ab$ is a crystal, then there are no other pairs of positive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and $B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.

Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $z\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(n) : n\in\mathbb{Z}\}$ such that $z=a+b$?

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).

The Jacobian Conjecture: any regular function (i.e. vector valued polynomial function from) kⁿ → kᵐ whose Jacobian is a non-zero constant has an inverse that is given by a regular function, where k is a field of characteristic 0

Benchmark open subproblem: existence of a SIC-POVM in dimension $58$.

The idempotent conjecture*

If G is torsion-free, then K[G] has no non-trivial idempotents.

Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of subgroups $G_1, \dots, G_k$ of $G$.

Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the indices $[G:G_1], \dots, [G:G_k]$ cannot be distinct.

Equivalent p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on any connected finite-dimensional topological manifold. By the Gleason–Yamabe theorem, this is equivalent to hilbert_smith_conjecture.

Lower bound for $c(p)$ for $1 < p \le \infty$, improving the known value at $p = 2$ or $p = \infty$.