17 problems

Let $X$ be a quasi-projective surface. For each $n$, let $H^*(X^{[n]})$ be expressed in the Heisenberg monomial basis $(\mathfrak h_\mu(n))$. Constant Conjecture. The structure con…

Let $S$ be a (quasi-)projective surface, and let ${\cal H}(S)$ be the direct sum of the homology groups of the Hilbert schemes $S^{[n]}$ of $n$ points on $S$. Vafa–Witten's conject…

Let $X$ be a smooth quasi-projective surface, and write … where $X^{[n]}$ is the Hilbert scheme of $n$ points on $X$. Heisenberg algebra conjecture. There exists a Heisenberg algeb…

Irreducibility for general $m$. For all $m\geq 3$, the equivalence of Theorem$\textup{}$ extends to ${\mathcal{H}_m}$ with multiple cocycle components…

Decategorification isomorphism conjecture. The injective algebra homomorphism above is always an isomorphism.

Grojnowski's conjecture. The Grojnowski–Nakajima result should hold for $X$ of any dimension if one replaces $X^{[n]}$ by the symmetric quotient orbifold $X^n/S_n$ and replaces rat…

Symmetric-power criterion conjecture. If the canonical morphism above is an isomorphism, then $\pi$ is an isomorphism.

Converse categorification conjecture. If $\bicat{F}_{{\cat V}}$ categorifies $F_{{\cat V}}$, then $\pi$ is an isomorphism.

Quantum Heisenberg categorification conjecture. There is an isomorphism

Let $H_{\mathbb{Z}}$ be the graded Heisenberg algebra with $\deg b_m=m$ and $\deg a_m=-m$ for $m\in\mathbb{Z}_{>0}$, and let $\dot{H}_{\mathbb{Z}}$ be the preadditive category whos…

The Heisenberg algebra inclusion conjecture. This injection is an isomorphism:

The main conjecture. (a) The set

Let $f^{(\kappa)}(x_1,\dots,x_\ell)$ be the $\kappa$th generator of the invariant polynomial ring $S(\mathfrak h)^W$, and let $M_0(\boldsymbol a)$ be the module in the source. Let…

Let $Q$ be a quiver with one vertex and $m$ arrows. For even $m$, let $Cl_c$ denote the infinite Clifford algebra with generators $\xi_n^{+}$ and $\xi_n^{-}$ for…

Let $S$ be a (quasi-)projective surface, and let $S^{[N]}$ denote the Hilbert scheme of $N$ points on $S$. Write ${\cal H}(S)$ for the direct sum of the homology groups of these Hi…

For each $n\geq 1$, let $R_0$ denote the remainder of the element $D_0$ in the construction of the minimal relation for ${\mathcal H}(n)^{O(n)}$. Nonvanishing remainder conjecture.…

Let ${\mathcal H}(n)$ be the rank $n$ Heisenberg vertex algebra, with automorphism group $O(n)$, and let ${\mathcal W}(2,4,6,\dots,n^2+3n)$ denote a ${\mathcal W}$-algebra of the i…