43 problems

Let $n$ be a positive integer, let $g\in L^{2}(\mathbb{R}^{n})$ be nonzero, and for $x,\omega\in\mathbb{R}^{n}$ let $T_x$ and $M_\omega$ denote the translation and modulation opera…

Define the multiple zeta values by … for $s_1\geq 2$ and $s_2,\ldots,s_p\geq 1$. Selected-value independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, $\zeta(2,3)$…

Define the multiple zeta values by … for $s_1\geq 2$ and $s_2,\ldots,s_p\geq 1$. Low-depth independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, and…

Let $\mathcal{A}$ be a cluster algebra over ground ring $\mathbb{Z}\mathbb{P}$. A cluster monomial is a monomial in cluster variables belonging to one cluster. Cluster-monomial lin…

Witness-elimination conjecture. With

Linear-independence conjecture. Fix a rectangle $R$. The products

Atiyah's conjecture. The polynomials $p_1,\ldots,p_N$ are linearly independent. Equivalently, their $N\times N$ coefficient matrix is nonsingular. Atiyah's conjecture is known for…

Linear Independence Conjecture. This set is linearly independent over $\mathbb{Q}$. The conjecture is used in the paper's large-deviation arguments for the summatory functions unde…

Weak mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

Canonical splitting conjecture. There is a canonical direct-sum decomposition

Mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

Chowla–Milnor conjecture. The set $\mathfrak{L}$ is linearly independent over $\mathbb{Q}$. The conjecture is important because its consequences include conditional irrationality r…

Let $d$, $k$, and $s$ be positive integers, and let $p_1,\ldots,p_k\in\mathbb{R}[x_1,\ldots,x_d]$ be homogeneous polynomials of the same degree $s$, no two of which are linearly de…

Equivalent Chowla–Milnor formulation. One has

Full-dimension conjecture. One has

Number-field linear independence conjecture. The numbers

Chatterjee–Garg conjecture. The numbers

Let $p$ be a prime, let $k>1$ be an integer, and let the Hurwitz zeta function be … for $0<x\le1$. Milnor's conjecture. The real numbers … are linearly independent over…

Let $q$ and $k$ be positive integers greater than one. Consider the Hurwitz zeta values $\zeta(k,a/q)$ for integers $a$ satisfying $1\le a\le q$ and $(a,q)=1$. Milnor's conjecture.…

Let $\Lambda\subset\mathbb{R}^{2}$ be a fixed finite set, let $f\in L^{2}(\mathbb{R})$ be nonzero, and write … Here $\mathcal{G}(f,\Lambda)$ is the associated finite Gabor system.…

Let $d\geq 3$ be an integer, let $|x\rangle$ be a vector on the unit sphere in $\mathbf R^{d-1}$, and construct matrices $A_m$ using $V_m=2|x\rangle\langle x|-I_{d-1}$ for every…

Let $J_{\mathfrak{D}}(1;{\rm odd},{\rm odd})$ denote the regularized double zeta values whose first argument is $1$ and whose remaining arguments are odd. Linear independence conje…

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Write … where the union is a multiset. LI− conjecture. The multiset…

Let $K=\mathbb{F}_q(t)$ be a rational function field, and let $\zeta(\mathbf{s})$ denote the multiple zeta values at positive-integer tuples $\mathbf{s}$. Thakur's linear independe…

Let $\operatorname{SLs}(k_{1},\dots,k_{n})$ denote a shifted log-sine integral, with $m,n\geq 0$ and $k_i\geq 2$. Linear-independence conjecture. The elements of … are $\mathbb{Q}$…