56 problems

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

Let $U_{q,p}(\widehat{\mathfrak{sl}_2})$ be the elliptic algebra introduced in the paper, with screening currents and representation spaces as constructed there. The q-deformed ext…

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…

Let $\tilde{\cal F}^{PF}_{J,M}$ be the restricted $q$-parafermion Fock module, and consider the complex … Here $Q^{[s]}_{J+1}$ denotes the alternating screening operators described…

The paper considers nonlinear differential equations, infinite-order linear differential equations, finite-difference equations, and q-differential equations, together with their q…

q-Meixner conjecture. The Bernoulli distribution should be $q$-Meixner for every $q$ with

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

Lenart's conjecture. The map $f^q_r$ has the property asserted in the continuation of the conjecture in the source.

Jorgensen–Schmitt–Werner conjecture. For any $|q_{ij}|<1$, the algebra $\mathcal{O}_n^{q_{ij}}$ is the Cuntz–Toeplitz algebra. This is the limiting case obtained by setting the dia…

Let $d$ be a positive integer, and let $a_i,a_i^*$, $i=1,\dots,d$, generate a $C^*$-algebra subject to … where $q_i\in(-1,1)$ and $|q_{ij}|\leq 1$. Jorgensen–Schmitt–Werner conject…

Let $\mathcal{A}^k_{k+s,k+r}$ be the vertex operator algebra generated by the currents $e(z)$ and $f(z)$, with $s$ and $r$ parameters and $k$ the representation level. A q-deformat…

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…

The factorisation … leads to bicrossproduct quantum groups. Bicrossproduct contraction conjecture. The resulting bicrossproducts are all nontrivially isomorphic to certain contract…

Regularity conjecture. Performing this singular gauge transformation gives

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…

Fix a Markov number $cell$ with $cell\geq 5$, and consider either of the two classical Markov-tree branches of triples $(cell,b,c)$ with $cell$ fixed. Let $m_{cell}(q)$ be the mirr…

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 universal radius-of-convergence conjecture. For every positive real number $x$,

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…