42 problems

For every complex simple Lie algebra $\mathfrak{g}$, with $\rho$ the half-sum of its positive roots, and every dominant integral weight $\lambda\in P_+$, the irreducible representa…

Let $a_n$ be the number of triangulations of a fixed convex $n$-gon such that at least six triangles meet at each internal vertex. Let…

Let $\mathfrak g$ be a finite-dimensional Lie algebra, let $n$ be a positive integer, and let $x_1,\ldots,x_n\in\mathfrak g$. Define … For each $i$, let $\partial_{x_i}F_i$ be the…

Let $\alpha_{ij}=e_i-e_j$ be the roots of the type $A$ root system. An alcoved polytope is a polytope whose faces lie in hyperplanes $\langle\alpha_{ij},\tau\rangle=m$ for…

Degree–singularity–root conjecture. The degree $\eta(w_*)$ of $p(q)$ is the multiplicity $d$ at $p_o$, where $d$ is given by the minimal degree of the product of the $\tau$-functio…

Poisson vector bundle conjecture. (1) Every homogeneous Hermitian Poisson vector bundle over ${\mathfrak g}^*$ is Poisson-trivial. (2) There is a one-to-one correspondence between…

Irrational-angle counterexample conjecture. It is impossible to find a counterexample satisfying $\varphi_0\notin 2\pi\mathbb{Q}$.

Generalized Duistermaat–Heckman conjecture. An analogous correlation-function formula should hold for every compact group, with the triangular integration domain identified with…

Let $X$ and $Y$ be Dynkin diagrams of ADET type, with Cartan matrices $\mathcal{C}(X)$ and $\mathcal{C}(Y)$, ranks $r(X)$ and $r(Y)$, and Coxeter numbers $h(X)$ and $h(Y)$. Define…

Let $\mathfrak{g}$ be a reductive Lie algebra. Choose a well-chosen $\mathbb{Q}$-factorial terminalization … and let $X_{\mathfrak{g},\mathrm{Losev}}$ be the intermediate symplecti…

Theta–$\mathfrak{sl}_3$ identification conjecture. Up to conventions and normalizations,

Let $F$ be a local or finite field, let $\underline G$ be a split semisimple group with Lie algebra $\mathfrak g$, and let $\Omega\subset\mathfrak g$ be a conjugacy class with clos…

Let $h:[n]\to[n]$ be a Hessenberg function, let $\mathsf{x}\in\mathfrak{gl}_n(\mathbb{C})$, and let $\varphi_{\mathsf{x},h}$ be the one-parameter family constructed from the associ…

Finkelberg–Fourier–Littelmann conjecture. For every birational sequence $S$ and monomial ordering $>$, the monoid $\Gamma(S,>,\lambda)$ is finitely generated.

Weinstein's conjecture. The dual $\mathfrak{g}^*$ admits a tangential star product if and only if there is a flat torsion-free affine connection on a neighborhood of the identity i…

Let $G$ be the point-symmetry pseudogroup under consideration, with splitting $G=G^{\rm ess}\ltimes G^{\rm lin}$ and Lie algebras $\mathfrak g$ and $\mathfrak g^{\rm ess}$. For…

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and let the statements of Theorems 1 and 2 of the paper be understood in their previously established generality. Exception…

Let $\mathfrak{g}$ be any simple real Lie algebra of rank $r$, and let $\lambda\in Q\cap\mathfrak{h}^+$ be a dominant weight, written in the basis of fundamental weights as … Suppo…

Let $\mathfrak{g}$ be a Lie algebra, let $\operatorname{Hilb}(\mathfrak{g})$ be its generalized Hilbert scheme, and let $\operatorname{Hilb}_0(\mathfrak{g})$ be the zero-fiber of t…

Let $\mathfrak{g}$ be a semisimple Lie algebra, and suppose a smooth version of $\operatorname{Hilb}(\mathfrak{g})$ exists. For classical $\mathfrak{g}$, let $m$ be the dimension o…

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ denote its generalized Hilbert scheme. Generalized GIT quotient conjecture. There is a g…

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ be its generalized Hilbert scheme with Chow morphism $ch$ to $\mathfrak{h}^2/W$. Chow-re…

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$, and let $\lambda$ and $\mu$ be weights such that the multiplicity $m(\lambda,\mu)$ of $\mu$ in the irreducible representatio…

Malkoun's singular-value conjecture. For every $\mathbf{x}\in X$,

Folding conjecture for classical multiplicative Langlands duality. There is an equivalence of categories