Open-problem statements, with their sources attached.

The zero-divisor conjecture*

If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.

Four exponentials conjecture* Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental.

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.

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 $1, ..., n$.

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

The idempotent conjecture*

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

Are $e$ and $\pi$ algebraically independent?

The MLC conjecture, stating that the mandelbrot set is locally connected.

The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges. (Note that we 0-index the series below.)

The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function f : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that f = supᵢ infⱼ(gᵢⱼ).

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

Is $e + \pi$ irrational?

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.

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

Is $e \pi$ irrational?

The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set.

Černý Conjecture*: Every synchronizing DFA with $n$ states admits a synchronizing word of length at most $(n - 1)^2$.

The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_n(I) is a nil ideal of the matrix ring M_n(R).