17 problems

Let $\pi$ and $\pi'$ be irreducible cuspidal automorphic representations of $\operatorname{GL}(m)$ and $\operatorname{GL}(m')$ over $\mathbb{Q}$, respectively, both with unitary ce…

Let $\mathcal S$ be the Selberg class. Let $F,G\in\mathcal S$ be primitive functions with Dirichlet coefficients $a(n)$ and $b(n)$, respectively, and define $\delta_{F,G}=1$ if…

Let $X=\mathbb{C}^n$ be equipped with an arbitrary norm. An operator $T\in B(X)$ satisfies the Bhatia–Šemrl property if, for every $A\in B(X)$ with $T\perp_B A$, there exists a uni…

Cauchy–Schwarz conjecture. The Cauchy–Schwarz inequality should hold in every complex normed space. The inequality is established in the paper for some classes, including complex r…

Large-cardinal dimension-range conjecture. The appearance of large cardinal values in the functional representation of the dimension range of an espalier should be closely related…

Generalized orthogonality conjecture. The correct notion of orthogonality is $\operatorname{NewOrtho}$, rather than $\operatorname{Ortho}$.

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 $n$ be a positive integer, let $\boldsymbol{x}_n,\boldsymbol{x}'_n$ be position variables, and let $\boldsymbol{\lambda}_n,\boldsymbol{\lambda}'_n$ be spectral parameters. Deno…

Ahmadi–Mohammadi conjectures. The inequalities in the displayed bounds for ${\mathcal S}_{3,2}(q,n,{\mathcal B})$ for odd $q$ and for ${\mathcal S}_{3,2}(2,n,\cdot)$ could be repla…

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

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

Scalar norm-minimality conjecture. The condition

Let $X$ be a finite-dimensional real smooth normed linear space with $\dim X\geq 3$. A finite set $\{x_1,\ldots,x_n\}$ is strongly orthonormal in the sense of Birkhoff–James if, fo…

Orthogonality-language conjecture. The answer is conjectured to be affirmative: Euclidean geometry can be expressed using $k$-subspaces as individuals and some of the introduced or…

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.