32 problems

Let $\mathbf{v}$ be a dimension vector, let $\GL_{\mathbf{v}}$ be the corresponding product of general linear groups, and let $\mathbf{N}_{\mathbf{v}}$ be the associated representa…

Let $G$ be a reductive group, let $\mathbf{N}$ be a representation, and let $\mathcal{M}(G,\mathbf{N})$ and $\mathcal{M}_{\mathsf{Higgs}}(G,\mathbf{N})$ denote the associated Coulo…

Let $Q=({\mathbb I},\Omega)$ be an ADE quiver, let $(V_i;W_i)_{i\in{\mathbb I}}$ be a representation of its framed double quiver, and impose the stated Satake-diagram symmetry and…

Let $X$ and $X^!$ be symplectically dual varieties arising respectively from Coulomb-branch and Higgs-branch constructions for the gauge theory $(G,N)$. Let $R$ be the coefficient…

Let $\Phi^{-,\mathrm{re}}_\mu$ be the relevant set of negative real roots, let $\langle\alpha,\mu\rangle$ denote the root–coweight pairing, and let $L$ be the unique irreducible ob…

Let $G$ act on ${\bf N}$, let $\nu\colon\mathbb{C}^{\times}\to\mathsf{A}$ define a partial resolution $\widetilde{\mathcal{M}}_{C,\nu}\to\mathcal{M}_C$, and let…

Let $\mathcal{M}^{!}=\overline{\mathcal{W}}^{\lambda}_{\mu}$ be the 3d mirror dual of a zero-dimensional quiver variety, with unique torus fixed point $p^{!}$, and set…

Homological Hikita conjecture. There is an isomorphism of graded $\mathbb C[(\mathfrak t_\bv/S_\bv)\times\ff]\otimes\mathbb C[\hbar]$-algebras

K-theoretic Hikita conjecture. There is an isomorphism of algebras over $\mathbb C[(T_{\bf{v}} / S_\bv) \times F] \otimes \mathbb C[q^{\pm 1}]$

A generalized quiver is a triple $(V,A,K)$, where $V$ is finite, $A:V\times V\to\mathbb{Z}$, and $K:V\to\mathbb{Z}_{>0}$, satisfying the diagonal and symmetrizability conditions st…

Let $Q$ be a good 3d $\mathcal{N}=4$ quiver theory, let $\mathcal{M}_C(Q)$ denote its Coulomb branch, and let $\mathcal{L}$ be the set of decay and fission products of $Q$, ordered…

Let $G$ be a simple group, let $g$ be the genus of a Riemann surface, and let $e^{(1)},\dots,e^{(p)}$ be nilpotent labels for its punctures. Write $\mathcal{M}(G^{\vee},V)$ for the…

Purity conjecture. The rational cohomology of every resolved Coulomb branch associated with connected semisimple groups, in the sense of Braverman–Finkelberg–Nakajima, has pure Hod…

Let $V_i$ and $V_j$ be finite-dimensional vector spaces, and let $\operatorname{GL}(V_i)$ and $\operatorname{GL}(V_j)$ act on the Hamiltonian spaces $\mathbf M_{\medcirc}(V_i,V_j)$…

Let $Y$ be a Hamiltonian $G$-manifold with maximal torus $T$, Weyl group $W$, and root system. Let $(Y^\vee,f)$ be its two-dimensional mirror Landau–Ginzburg model, let…

Let $G$ be a reductive group, let $\mathcal A_G=\bigoplus_{d=0}^{\infty}{}_0\mathcal A_d$ be the graded Coulomb branch algebra, and for $\gamma\in\mathfrak g$ define … Let…

Let $\Gamma$ be an ADE root system, let $Q$ be an orientation of its Dynkin diagram, and let $\alpha=\sum_{i\in Q_0}n_i\alpha_i$ be a positive root. Set…

The construction associates to each $gammabin g(cK)$ a {cC}^-equivariant quasi-coherent sheaf $cFgamma$ on $cmathfrak C{ccheck G}$. Coherence conjecture. For any regular semi…

Let $I=(I_0,I_1)$ be a finite quiver with dimension vector ${\bf v}$ and framing ${\bf w}$, let $A$ be the natural torus acting on $\mathfrak{M}({\bf v},{\bf w})$, let $G_{\bf v}$…

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 $\mathcal M^\nu(\lambda,\mu)$ be the deformation associated with a decomposition $W=W^1\oplus W^2$, let $\lambda^1$ and $\lambda^2$ be the corresponding dominant coweights, and…

Let $W=W^1\oplus W^2$ be a decomposition inducing dominant coweights $\lambda^1$ and $\lambda^2$, let $\mathcal M^\nu(\lambda,\mu)$ be the corresponding deformation of the Coulomb…

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi$ be the one-parameter subgroup defined by a character of the gauge group, and l…

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi\colon\mathbb C^\times\to\pi_1(\mathbf G)^\wedge$ be the one-parameter subgroup…

Let $G_c$ be the compact form of a gauge group, and let $mu$ and $\lambda$ be coweights with $\lambda$ dominant. For dominant $nu$ satisfying $mu\leqnu\leqlambda$, let…