10 problems

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…

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 $X=T^*Gr(k,n)$, 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…

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

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 $X$ 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…