7 problems

Hilbert-space characterization conjecture. The following conditions are equivalent: $\mathbb{X}$ is a Hilbert space; and a nonempty set $A\subset S_{\mathbb{X}}$ is a minimal…

Let $A\subseteq S^2$ be a measurable subset containing no two orthogonal directions. Its measure is normalized by $|S^2|=1$. Kalai–Wilson's conjecture. The maximal measure of such…

Let $\pi$ and $\pi'$ be self-dual unitary cuspidal automorphic representations of $\operatorname{GL}_m(\mathbb A)$ and $\operatorname{GL}_{m'}(\mathbb A)$, respectively, and let…

Spin Whittaker orthogonality conjecture. For all $\underline{L}_N,\underline{L}'_Ninmathpzc{W}_N$,

Spin q-Whittaker orthogonality conjecture. For all signatures $\lambda,mu$,

Congruence twin conjecture. Suppose that the dimension $n\geq 5$ is odd and $d\not\equiv n\pmod{8}$. Then every vector of length $\sqrt{d}$ has a twin.

Odd-dimensional twin conjecture. Suppose that the dimension $n\geq 5$ is odd. Then every non-odd vector has a twin.