25 problems

Let $Q_n^{+[t]}$ be the maps in the complex associated with the screening charge $Q_n^+$, and let ${\cal H}_{J;n',n}$ denote the proposed irreducible highest weight representation…

Uniqueness conjecture. At an arbitrary value of $p$, this solution of the system is unique.

Representation isomorphism conjecture. The corresponding representations of $\mathcal{Z}_q$ and $\widetilde{\mathcal{H}_{GT}}$ on quantum field theories on $S^{4,\theta}$ are isomo…

Consider triples in the mirror deformation of the Markov tree, with each triple consisting of mirror deformed Markov numbers and with degree taken in the deformation parameter $q$.…

Let $m_n$ denote the mirror deformed Markov numbers. Positivity conjecture. Every coefficient of every $m_n$ is positive. The claim is based on experimental computations and is sta…

Fix a Markov number $\ell\geq 5$. Let $m_\ell(q)$ be a mirror deformation of $\ell$, and let $(p_n)_{n\geq 0}$ be the sequence of polynomials on a mirror Markov-tree branch fixing…

Let $\Delta=([\ell],\mathcal{F})$ be a simplicial complex on $\ell$ vertices, and let $\mathcal{S}_\Delta^q$ be its generalized $q$-deformation. Let $G$ be the underlying graph of…

Let $G=([[200~\ell], E)$ be a simple graph on $\ell$ vertices, and let $\mathcal{A}_G^q$ be its $q$-deformation of graphic arrangement. Write $\chi(\mathcal{A}_G^q,t)$ for its char…

Assume $q\neq0$. Let $\operatorname{\mathcal{H}}_n^{\operatorname{aff}}$ be the affine Hecke algebra containing $\operatorname{\mathcal{H}}_n$ and the Laurent polynomial ring in…

The case $\ell\geq n+2$. The sequence $\Delta_j^{(n+2)}$ is $2n(n+1)$-periodic when $n$ is even and $2n(n+1)$-antiperiodic, hence $4n(n+1)$-periodic, when $n$ is odd, and its terms…

For a sequence of positive integers $a_1,\ldots,a_{2m}$, let $M(a_1,\ldots,a_{2m})$ be the corresponding matrix and call a polynomial $f(q)=\sum_{i=0}^n a_iq^i$ unimodal if…

Let $G$ be a graph, let $\mathcal{A}_G^q$ be its $q$-deformed graphical arrangement, let $\chi(\mathcal{A}_G^q,t)$ denote the characteristic polynomial of this arrangement, and let…

Consider the specializations … Let $\mu_{n,k}^b$ be the corresponding measure, with $q>0$, $b\neq 0$, and let $\rho(t)$ and $\Omega(t)$ denote its limiting density and limit shape.…

Let $p$ be an odd prime integer, and let $a,b$ be positive integers coprime to $p$. The associated $q$-deformed rational-number polynomials are denoted by $\mathcal{S}_{a/p}(q)$ an…

Let $L_q$ be the proper subset of $\mathbb{H}$ covered by the $q$-deformed Farey tessellation. Its boundary satisfies $\partial\overline{L}_q=\mathbb{Q}_q\sqcup\mathbb{I}_q$, where…

Let $\left[\frac{r}{s}\right]_{q}=\frac{\mathcal{R}(q)}{\mathcal{S}(q)}$ be a $q$-rational, where $\mathcal{R}(q)$ and $\mathcal{S}(q)$ are its numerator and denominator polynomial…

Generic bilinear relation.

Algebraic-solution relation. For $s=\pm1$,

Bilinear equation. The function $\mathcal{T}(u,s;q|Z)$ satisfies

Bilinear relations. The $q$-deformed conformal blocks satisfy

Let $R$ be a smooth $\mathbb Z$-algebra, and let a framing $\square$ define the category of finite projective $R[[q-1]]$-modules equipped with flat $q$-connections with respect to…

Let $R$ be a smooth $\mathbb Z$-algebra with $p$-adic completion $\widehat R$. The Frobenius conjecture. The $p$-adic completion $\widehat{q\!\operatorname-\!\Omega_R}$ admits a…

Let $R$ be a smooth $\mathbb Z$-algebra, and let $\square:\mathbb Z[T_1,\ldots,T_d]\to R$ be an étale framing. The framed complex…

Let $m,n\in\mathbb{N}$, let $p$ divide $m$, and let $G(m,p,n)$ be the complex reflection group contained in $G(m,n)$. Its $q$-harmonic space is defined using the $q$-deformed opera…

Let $W=G(m,n)$, let $\ell\in\mathbb{N}$, and let $q$ be formal. Define $\mathcal{D}W^{(\ell)}_q$ as the space of polynomials annihilated by all operators $D_{q,\mathbf{d}}$ for whi…