29 problems

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

Let $Z=Z_1\cup\cdots\cup Z_s$ be a union of curves, let $\langle Z\rangle$ be the relevant semigroup of classes, and let $\overline{\mathfrak a}_\gamma$ be the amalgamation map in…

Let $S$ be the surface in the preceding setup, let $C_i$ be one of the distinguished curves, and let $[\mathfrak Y_{i,(d)}]$ and $[\mathfrak Z_{i,\ell}]$ denote the corresponding d…

Let $X$ be a smooth surface, let $Z_1$ and $Z_2$ be curves with $Z=Z_1\cup Z_2$, and let $\mathbf{HA}^T_{X,Z}$ denote the corresponding $T$-equivariant cohomological Hall algebra.…

BPS complex conjectures. The following assertions should hold:

Let $K>2$, and let $Y^{K}$ be the $C[Chbar_1,Chbar_2]$-linear algebra obtained by gluing the two copies of the double deformed current algebra $A^K$ constructed from observables on…

Let $X$ be a two-sided Hecke pattern of rank $r$. Write $H_*(X)$ for its Borel–Moore homology, and let $W_{\star,\star}(S)$ denote the algebra with the appropriate choices of indic…

Shuffle-algebra image conjecture.

Let $r,d\in\mathbf{Z}$ with $r>0$, and let $\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}}$ be the indicated Betti CoHA subalgebra object. Let…

Let $f\colon\mathcal{C}\to S$ be a family of smooth projective curves of genus $g$ over a smooth connected base $S$. Let $\mathfrak{M}_{r,d}^{\operatorname{Dol}}(\mathcal{C})$ be t…

Let $Q=(Q_0,Q_1)$ be a quiver, let $\lambda\in\mathbf{C}^{Q_0}$, and let $\Pi_Q^{\lambda}$ be the $\lambda$-deformed preprojective algebra. Write…

Let $(g,r,d)\in\mathbf{N}\times\mathbf{Z}_{\geqslant 1}\times\mathbf{Z}$ be a triple consisting of a genus $g$, a rank $r$, and a degree $d$. The polynomial…

Let $Q$ be the quiver, let $\mathbf{d}\in\mathbb{N}^{Q_0}$, and let $\mathfrak{N}_{\Pi_Q,\mathbf{d}}^{\mathcal{SSN}}$ be the seminilpotent stack, with irreducible components…

Let $X$ and $X'$ be the varieties in the induction diagram, with a map $g\relationships W\to X'$ and the induced map … where … Let…

Let $X$ be the local toric Calabi–Yau $3$-fold in the source, let $Y_\infty$ be the geometric Cartan doubled Yangian, and let $V:=H^*_c(\fM_\zeta^\chi,\varphi_W\mathbf{C})^\vee$ be…

Let $Q$ be the quiver associated with a noncommutative crepant resolution $A=\mathbb{C}\{Q,W\}$ of the threefold setup, let $C$ be the flopping curve, and let $\Phi(0,1)$ be its di…

The vertex-algebra existence conjecture. One can associate to $\mathcal{C}$ a class of vertex algebras parametrized by rays and by a choice of framing object; for almost all rays,…

Let $Q$ be a Dynkin quiver with vertex set $Q_0$, and let $\beta$ be a positive root of $Q$. For each vertex $i\in Q_0$, write $\alpha_{i,1},\ldots,\alpha_{i,\beta(i)}$ for the cor…

Let $\mathbb C$ be the configuration-space curve, let $\operatorname{Ran}(\mathbb C)$ be its Ran space, and for $x\in\operatorname{Ran}(\mathbb C)$ let…

Let $Q$ be a quiver, let $\mathbb{Y}_+$ be the positive half of the Maulik–Okounkov Yangian, let $K$ be the scalar extension used in the paper, and set…

Let $\mathcal M_{k,d,N,\alpha}$ be the relevant moduli space of quasi-maps, let $\mathcal P$ be the perverse sheaf lifting the Behrend-function data, and let…

Orientifold CoHA structure conjecture. The CoHA action map

Orientifold CoHA freeness conjecture. If $\mathcal{H}_Q$ is supercommutative, then the CoHA action map

The integrality conjecture.

Let … and let $u$ be a formal variable. Schiffmann's conjecture. There is a Lie algebra structure on … and a filtration $Y$ on $\mathcal{H}^{\operatorname{\mathtt{Betti}}}_g$ such…