20 problems

Let $m,n$ be coprime integers. Let $\mathcal{F}_u$ be the Fock module, and let $s_\lambda^{m/n-\epsilon}$ and $s_\lambda^{m/n+\epsilon}$ be the stable-envelope bases associated wit…

Elliptic quantum group and geometric representation conjectures. (1) Vertex operators of $U_{q,p}(\widehat{\mathfrak g})$ or $U_{t_1,t_2,p}(\mathfrak g_{tor})$ defined as intertwin…

Let $X$ be a type A Nakajima quiver variety with its natural torus action, let $p$ be a $\mathsf{T}$-fixed point, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. Qu…

Let $lambda$ be a partition in an $(n-k)\times k$ rectangle, let $\lambda^c$ be its complement, and let $\alpha$ be the automorphism of $\mathbb{C}(\hbar,a_1,\ldots,a_n)$ defined b…

Let $xi_{\ell,r,c}$ denote the integer in the reduced $Gr_2$-simplex in layer $\ell$, row $r$, and NE-SW diagonal $c$, and set $\xi_{\ell,r,c}=0$ when there is no entry in that pos…

Arrange the entries in each layer of the $Gr_2$-simplex as coefficients of a polynomial in $a,b,c$ according to the arrangement described in the paper. Divisibility conjecture. Eac…

For each $n\geq 2$, arrange the $dbinom{n}{2}$ integers obtained from the non-equivariant integrals of stable envelopes over $T^*Gr(2,n)$ into the $Gr_2$-simplex, with layers index…

Let $X=T^*Gr(k,n)$, let $\boldsymbol{a}=(a_1,\ldots,a_n)$ be the equivariant parameters, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. For a partition $lambda$ in…

Let $\mathrm{Hilb}^n(\mathbb{C}^2)$ denote the Hilbert scheme of $n$ points on $\mathbb{C}^2$, and let DAHA Verma modules be the corresponding representation-theoretic objects. Sta…

Let $U_q(\hat{\mathfrak{g}}_Q)$ be the quantum algebra constructed via the $K$-theoretic stable envelope, and let $U_q(\hat{\mathfrak{g}}_Q)$ denote the quantum affine algebra of t…

Let $\Lambda$ be the space of symmetric functions, and write the vectors $|\lambda\rangle_\infty$ in the bases of Macdonald symmetric functions $M_\mu$ and Schur functions $s_\mu$…

Okounkov's shift-operator R-matrix conjecture. In the stable basis, the operator $S_\mathsf{w}$ is conjugate to the R-matrix for

Okounkov's shift-operator conjecture. In the stable basis, the operator $S_\mathsf{w}$ is conjugate to

Okounkov's stable-basis R-matrix conjecture. In the stable basis, the operator $\mathsf{B}_\mathsf{w}$ is conjugate to the R-matrix for

3D-mirror conjecture. There exists a dual variety $\mathsf{X}^{!}$ such that the twisted elliptic stable envelopes of $\mathsf{X}$ and $\mathsf{X}^{!}$ coincide after this identifi…

Let $\DD$ be a brane diagram and let $f\in\Ch(\DD)^{\T}$ be a fixed point. For each tangent-weight term $\alpha_{x,y,k}\frac{x}{y}\mathbf h^k$, call it $f$-small when its restricti…

Let $X$ and $X'$ be 3d mirror varieties, and let the elliptic stable envelopes of $X$ and $X'$ be defined. Denote by “properly renormalized” the elliptic stable envelopes after the…

Let $X$ be a smooth symplectic variety with finite $X^{\mathsf{T}}$, and suppose that the elliptic stable envelope exists. For $4\lambda,\mu\in X^{\mathsf{T}}$, define … Here $a$ a…

Stable-envelope existence and uniqueness conjecture. There exist unique classes $(S_p)_{p\in\mathfrak M_1^T}$ in $H_{\dim\mathfrak Z}^T(\mathfrak M_1)$ such that, for every…

Let $U_{q,p}(\widehat{{\bf sl}}_N)$ be the dynamical elliptic quantum group used to construct elliptic weight functions of type $\widehat{{\bf sl}}_N$, and let elliptic stable enve…