16 problems

Reduced-coordinate conjecture. The variables $\mathbf{w}$ are cluster coordinates on $\mathcal{X}_{\mathrm{red}}$: every mutation in the $\mathbf{w}$ variables can be lifted to a s…

Let $G$ be a connected complex semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Let $(e,h,f)\in\mathfrak{g}^{\times 3}$ be a principal $\mathfrak{sl}_2$-triple, a…

The cotangent bundle T^*\bm\hat{\mathcal{T}}^n is obtained by Hamiltonian reduction from the hyperkähler punctual Hilbert scheme, with its given symplectic structure near the zer…

Let $\mathfrak{g}$ be a simple complex Lie algebra, let $G$ be the corresponding group, and let $\operatorname{Comm}(\mathfrak{g})$ be the subscheme of…

The Hamiltonian reduction framework is considered for geodesic motion on a symmetric space of negative curvature, with spinless Calogero models obtained in the cases classified usi…

Quantization-reduction conjecture. There should be an isomorphism of algebras

Let $(\sigma_1,\sigma_2,\sigma_3)$ be three fermionic screenings whose momenta satisfy … Let the commutant be the algebra of fields commuting with these screenings, and call it non…

Non-realizability conjecture. The closure $\overline{\mathcal{O}}_\textrm{min}^{E_6}$ is not isomorphic to any Hamiltonian reduction of the cotangent space $T^*V$ of a representati…

Monomial-reduction conjecture. There exists a mutation sequence

Let $\lambda=(\lambda_1,\dots,\lambda_n)\vdash N$, let $a$ be the unique integer with $\lambda_{a-1}=1$ and $\lambda_a>1$, and define … Set $m=\lambda_1+\dots+\lambda_a$, let…

Let $\mathcal{M}_*^{\mathrm{red}}$ and $\mathcal{M}_{**}^{\mathrm{red}}$ be the reduced spaces defined from the corresponding strata of the reduced system, and let a maximal symple…

Let $p>0$ be an integer divisible by the lacity $r^\vee$, let $\mathfrak{g}$ be the relevant Lie algebra, and let $\mathrm{H}^{\kappa^*\!\!}\mathcal{A}^{(p)}[\mathfrak{g},\infty]$…

Partial compactification conjecture. The variety $\overline{X}_{\tau}$ is the desired partial compactification of $X_{\tau}$. This construction is intended to provide a systematic…

Let $\mathcal{A}//\mathcal{P}$ denote the space of parabolic connections with its infinitesimal higher-diffeomorphism action, and let $\operatorname{Symp}_0$ denote the correspondi…

Let $\mathcal{A}//\mathcal{P}$ be the space of parabolic connections, equipped with the infinitesimal action obtained by changing the line subbundle $L^*$. The acting infinitesimal…

Let $e_1,e_2\in\mathfrak{g}$ be nilpotent elements, and let $\mathfrak{m}_1,\mathfrak{m}_2\subseteq\mathfrak{g}$ be the subalgebras constructed for a covering pair of nilpotent orb…