36 problems

Yvonne's conjecture. The matrix $abla^{+}(q)$ coincides with the $q$-analogue of the decomposition matrices of these cyclotomic Schur algebras.

Let $\mathcal{PH}_\alpha^2(\mathbb{C}^n)$ be the pluriharmonic Fock space, let $\mu$ be a positive Borel measure, and let $T_\mu$ be the associated Toeplitz operator. For $r>0$, wr…

HRT conjecture. The functions

Hankel-product boundedness conjecture. The Hankel product $H_{\overline{f}}^{*}H_{\overline{g}}$ is bounded if and only if $D(f,g)$ is bounded.

Let $\Pi$ be the indexing set of partitions, let $|\emptyset\rangle$ be the empty-partition vector in the $q$-deformed Fock space, let $b_\lambda$ denote its canonical basis elemen…

Rouquier's conjecture. For an arbitrary multi charge, the multiplicity of a simple module in a standard module in $\mathcal{O}_{\boldsymbol{s}}(\ell,1,m)$ equals the corresponding…

Let $n$ be a positive integer, let $p=2n+1$ be a sufficiently large prime, and consider the level-one $q$-Fock space for the twisted affine algebra $U_q(A^{(2)}_{2n})$, together wi…

Let $m$ be a degree, let $n$ be a positive integer, and let $H_m$ be the type $A$ Hecke algebra specialized at an $n$th root of unity. Consider the degree $m$ part of the transitio…

Let $\mathbf{s}_l \in \mathbb{Z}^l(s)$, and let $\mathbf{b}_{\lambda_l,\mathbf{s}_l}$ be the canonical-basis element indexed by $\lambda_l$ in the crystal component…

Sharp stabilization-bound conjecture. Theorem 3.3 remains valid when its bound $N$ is replaced by $N'$.

Refined charge-gap conjecture. The statement that $\dot{w}+\dot{\alpha}_i$ is not a weight of $\boldsymbol{\operatorname{F}}_p[\boldsymbol s_n]^{\bullet}$ should hold whenever

Uglov's conjecture. The transition matrix of the homogeneous component of degree $m$ of the canonical basis of $\boldsymbol{\operatorname{F}}_q[\boldsymbol s_l]$ is equal to the de…

Arbitrary-charge Fock-space conjecture. For arbitrary $s_i$, the multiplicity of a simple module in a standard module in ${\mathcal O}$ equals the corresponding coefficient of the…

HRT conjecture. The functions $g_k$, $1\le k\le N$, are linearly independent in $L^2(\mathbb R,\mathrm{d}x)$. This is a central open problem in time-frequency analysis, also called…

Fixed-point conjecture. If for any $m>0$ we have $B_mf=f$, then $f$ is harmonic.

Let $F^p_\alpha$ be the Fock space on ${\mathbb C}^n$, let $L^p({\mathbb C}^n,dv)$ be Lebesgue $L^p$ space, and let $d_\alpha(z,w)$ be the distance function associated with the Foc…

Let $H({\mathbb C}^n)$ denote the entire functions on ${\mathbb C}^n$, let $F^p_\alpha$ be the Fock space, let $L^p_\alpha$ denote the corresponding weighted $L^p$ space, and let…

Let $alpha$ be a closed holomorphic one-form on $X$ with a unique Morse vacuum, defining a cyclic holomorphic cover $tilde{X}$ with deck group $Gamma=\mathbb{Z}langlegammarangle$.…

Let $\bm\lambda,\bm\mu\in\mathcal{P}^r$ lie in a block $B$ of $\mathcal{H}_{r,n}$, with $\bm\mu$ $e$-multiregular. Obtain $\bm\lambda^{+\varnothing}$ and $\bm\mu^{+\varnothing}$ fr…

The global derivative contraction conjecture. For every $f\in\mathcal{F}_\alpha^2$,

Let $\mathcal{F}_\alpha^2$ be the Fock space, let $f^{(n)}$ denote the $n$th derivative of $f$, let $\mu_{n,\alpha}$ be the measure used to define the weighted derivative norm, and…

Extended implementation conjecture. A statement similar to the bosonic and fermionic extended implementation theorems can be established for generic $v^*v$.

Let $l$ denote the level of the Fock space, and let $G(\lambda,\mathbf{s})$ be a canonical basis element. For $l=2$, the canonical basis elements are monomial vectors. Monomiality…

Let $F^p_\phi$ be a doubling Fock space over $\mathbb{C}^n$, with $n>1$, and let $T_f$ be a Toeplitz operator on this space. The paper's main Fredholm result asserts a characteriza…

Higher-dimensional jet interpolation conjecture. The following conditions are equivalent: