Open-problem statements, with their sources attached.

The number $\pi$ is normal in base $10$, so every block of $k$ decimal digits has limiting frequency $10^{-k}$.

The sequence will eventually reach $1$.

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative.

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

Benchmark open subproblem: existence of a SIC-POVM in dimension $56$.

The zero-divisor conjecture*

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

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

If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)

The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.

Hilbert–Smith conjecture*: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group.

Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.

For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then $\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$

What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?

Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?

A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii is $≤ 2$?

Let $f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x]$ with $|z_i| < 1$ for all $i$.

Conjecture: Must there always exist a path of length less than 2 in $\{ z \in \mathbb{C} \mid |f(z)| < 1 \}$ which connects two of the roots of $f$?

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.

(A) Existence and smoothness of Navier–Stokes solutions on ℝ³.

Conjecture 1.3* (the $\times p, \times q$ conjecture): the only atomless Borel probability measure on $\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.

Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$.

Then, for all sufficiently large $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.

Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?