10 problems

Let $\mathbf{j}$ and $\mathbf{j}'$ be two reduced decompositions of the longest element of the Weyl group of $\mathfrak{sl}_{n+1}$. For each reduced decomposition, consider the ass…

Dolbeault–Dirac spectral triple conjecture. The point spectrum of $D_{\overline{\partial}}$ has finite multiplicity and tends to infinity. Consequently,

Positive definiteness conjecture. For $q\in\mathbb{R}\setminus\{-1,0\}$, the pair $(\Omega^{(\bullet,\bullet)},\kappa)$ is a positive definite covariant Kähler structure.

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold of weak Gelfand type, and let $(\Omega^{(bullet,bullet)},\kappa)$ be its covariant Kähler structure, unique up to r…

Let $\mathcal{O}_q(\mathbb{OP}^2)$ be the quantum Cayley plane and $\mathcal{O}_q(\mathbf{F})$ the quantum Freudenthal variety, each equipped with its Heckenberger--Kolb calculus.…

Let $Z_S$ be the additive monoid of highest-weight-vector weights for an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$. Say that $Z_S$ is generated by $r$ elements if it…

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold, with its Heckenberger--Kolb calculus $\Omega^{(bullet,bullet)}$. A covariant Kähler structure consists of the pai…

Let ${\mathbb C}_q[G_0/L_0]$ be an irreducible quantum flag manifold. Let $(\Omega^{(\bullet,\bullet)},\kappa)$ consist of the Heckenberger–Kolb complex structure and a left-coinva…

Let $\mathbb{C}_q[G/L_S]$ be an irreducible quantum flag manifold, and let $\Omega^{(0,\bullet)}$ be the subcomplex of its Heckenberger--Kolb calculus. Denote by…

Let $\mathbb{C}_q[G_0/L_0]$ be an irreducible quantum flag manifold, with its Heckenberger--Kolb differential calculus $\Omega^{(\bullet,\bullet)}$ and the left-coinvariant form…