Open-problem statements, with their sources attached.

Is $e ^ e$ irrational?

The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all parameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n. Note that we need to require 2 ≤ n because the conjecture is trivially false for n = 1.

Fuglede's conjecture* in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_2(I) is a nil ideal of the matrix ring M_2(R).

Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.

Is $\pi ^ e$ irrational?

The boundary of the Mandelbrot set is conjectured to have zero area.

Fuglede's conjecture* in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.

The Köthe conjecture: for any positive integer n, the Köthe radical of R is the matrix ideal M_2(Nil*(R)).

In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value of the Landau constant.

Is $\pi ^ \pi$ irrational?

The boundary of any Multibrot set is conjectured to have zero area. Note that we don't need to exclude the trivial cases n = 0 and n = 1 because the conjecture holds for them.

The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.

Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$. Is $G$ necessarily finite?

Brennan's conjecture, part 1: $B(-2) = 1$.

Is $\ln(\pi)$ irrational?

If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least $e^{\sqrt n}$ in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.

Determine the value of the Busy Beaver function at n = 6.

Brennan's conjecture, part 2: $B_b(-2) = 1$.

Is the Euler-Mascheroni constant $\gamma$ irrational?