35 problems

Let $Y\to X$ and $Y^{\vee}\to X^{\vee}$ be a pair of dual symplectic resolutions with Hamiltonian torus actions. Fix a generic embedding of $\mathbb{C}^{\times}$ into the torus tha…

Let $\mathfrak{M}_0$ be a Nakajima quiver variety resolved by a smooth Nakajima quiver variety $\mathfrak{M}\to\mathfrak{M}_0$. Let $S_{\mathfrak{M}_0}$ be the torus acting on…

Let $X$ and $X^!$ be symplectic dual cones with Poisson brackets of degree two. Let $\operatorname{HP}_0(\mathcal{O}(X))$ denote the zeroth Poisson homology of the coordinate ring…

Let $G$ be a reductive group, let $\Gamma$ be a closed subset of its roots, and let $H_{\Gamma}$ be a maximal-rank subgroup of $G$ whose roots form $\Gamma$. Let $G^{\mathbb{L}}$ b…

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, let $G^{\mathbb{L}}$ be its Langlands dual with Lie algebra $\mathfrak{g}^{\mathbb{L}}$, let $U$ and $T^{\mathbb{L}}$…

Let $G$ be a reductive group, let $G^{\mathbb{L}}$ be its Langlands dual, let $\mathcal{I}\subset\Pi$ index a Levi subgroup $L_{\mathcal{I}}$, and let $U^{\mathbb{L}}$ be the relev…

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, let $G^{\mathbb{L}}$ be its Langlands dual with Lie algebra $\mathfrak{g}^{\mathbb{L}}$, and let $\mathfrak{A}$ and…

Let $X$ and $X^!$ be symplectically dual varieties arising from Coulomb-branch and resolved Higgs-branch constructions for the gauge theory $(G,N)$. Let $\zeta$ be the relevant roo…

Let $X$ and $X^!$ be symplectically dual, and let $\mathcal{A}$ be the universal quantization of $X$. The quantization exact sequence and the equivariant cohomology exact sequence…

Let $X$ and $X^!$ be symplectically dual conical Hamiltonian symplectic resolutions, with equivariant and Kähler roots, rings of differential operators, and associated $D$-modules…

Let $X$ be the nilpotent orbit considered in the paper, let $T$ be the torus used for its Hamiltonian reduction, and let $X^{!,\uparrow}$ be the corresponding BFN Coulomb branch, c…

Special-linear reduction conjecture. The Hikita conjecture holds for the pair $X\!/\!/\! /T$ and $X^{!,\uparrow}$. This conjecture proposes that the Hikita correspondence is preser…

The Hikita map is the map relating the cohomology of a symplectic resolution to the coordinate ring of the scheme-theoretic fixed points on its symplectic dual. A related dual pair…

Elliptic stable-basis duality conjecture. For any $p_1,p_2\in X^H$,

Let $G$ be a complex reductive group with Lie algebra $\fg$, let $G^{\vee}$ be its dual group with Lie algebra $\fg^\vee$, and let $\OO^\vee\subset\fg^\vee$ be a nilpotent orbit. L…

Let $I=(I_0,I_1)$ be a finite quiver with dimension vector ${\bf v}=(v_i)_{i\in I_0}$ and framing ${\bf w}=(w_i)_{i\in I_0}$. Let $A$ act on the Nakajima quiver variety…

Let $X\to X_0$ and $X^!\to X_0^!$ be symplectic dual conical symplectic resolutions, and let a torus $\mathbb{T}$ act on $X_0^!$. Denote by $H^*(X)$ the cohomology of $X$ and by th…

Let $X$ be a symplectic resolution and let $X^!$ be its symplectic dual. The quantum D-module of $X$ and the character D-module of $X^!$ are considered after a certain specialisati…

A symplectic resolution $X$ is equipped with an analogue of the BGG category $\cal{O}$. Symplectic duality conjecture. Symplectic resolutions should occur in pairs $X$ and $X^!$ su…

Let $\mathfrak{g}$ be a simply laced simple Lie algebra, let $\Gamma\subset SL(2,\mathbb{C})$ be the finite subgroup corresponding to $\mathfrak{g}$, and let…

Let $\mathfrak{g}$ be a Lie algebra, let a sheet be a stratum in $\mathfrak{g}$ associated with a nilpotent orbit, and distinguish special from non-special sheets and orbits. Let…

Let $G$ be a complex semisimple group, let $u$ be a dominant coweight, and let $mathcal{G}rmu^lambda$ be the transversal slice to the orbit $mathcal{G}r^mu$ inside the Schubert va…

Let $\mathcal T_{2d}$ be the common fully reduced two-dimensional theory obtained from the Higgs- and Coulomb-branch sigma models by sending the corresponding real mass or FI param…

Let $\mathcal O_H^{k_t,m_{\mathbb R}}$ be the relevant category associated to the Higgs branch, and suppose it admits an internal grading $\rho$ with homological grading $\eta$ suc…

Let $A$ be the quantized algebra of operators on the Higgs branch, with subalgebras $A_<$, $A_0$, and $A_>$ of negative, zero, and positive $U(1)_m$ charge. For a vacuum $D708$, le…