Open-problem statements, with their sources attached.

Benchmark open subproblem: existence of a SIC-POVM in dimension $75$.

Gottschalk's surjunctivity conjecture* (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.

$\Gamma(1/n)$ for n ≥ 2 is transcendental.

For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?

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

Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.

The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold. A topological space $X$ is an $n$-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X). The hypothesis [MetrizableSpace X] implies T2Space X so this does not appear in the conclusion.

Waldschmidt [Wal03] conjectured that a stronger result holds, namely that there exists a positive real number~$c$ such that $\lVert e^n \rVert > n^{-c}$ for every~$n \ge 2$. This is supported by metrical results [Kok45].

Note: the bound $n^{-c}$ equals $1$ when $n = 1$ for all $c$, while the distance to the nearest integer is always at most $1/2$, so the conjecture must start at $n \ge 2$.

The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals.

Do SIC-POVMs exist in every finite dimension?

Are there infinitely many binomial coefficients with deficiency 1?

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

Is it true that $f(n)\leq n^{o(1)}$?

Any T2, Toronto space is discrete.

Problem 10.4. Let $\xi$ be a non-zero real number and $\alpha > 1$ be a real number. The spectrum of the sequence $(\xi \alpha^n)_{n \ge 1}$ is at most countable. Posed by Mendès France [Men73].

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (\lambda(A)_i + \lambda(B)_{\sigma(i)})$? Here the numbers $\lambda(A)_i$ and $\lambda(B)_i$ are the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric group $S_n$.

Are there only finitely many binomial coefficients with deficiency > 1?

Inscribed square problem* Does every Jordan curve admit an inscribed square?

Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$?