28 problems

Motivic and $e$-polynomial mirror symmetry conjecture. The motives and $e$-polynomials of the character varieties $\mathcal{X}_G(\Gamma)$ and $\mathcal{X}_{^L G}(\Gamma)$ are equal…

Asymptotic ADHM generating-series conjecture. The generating series satisfies

Let a linear $\alpha_p$-action and a linear $\mathbb{Z}/p$-action be given on affine spaces, and consider their associated quotient singularities and stringy motivic invariants. Eq…

Let the orientation constructed by Joyce and Upmeier be the orientation used to define the motivic Donaldson–Thomas invariants in this paper, and let the orientation used by Behren…

Let $\mathbb k$ be a field, and let $C$ and $C'$ be smooth projective curves over $\mathbb k$ that are D-equivalent and whose geometric irreducible components have genus $1$. The a…

Let $c$ be the motivic birational invariant considered in the paper, and let $S$ and $S'$ be D-equivalent K-nef surfaces. The corresponding elements are the differences…

Let $K_0(\operatorname{Var}_k^{\dim})$ be the graded Grothendieck ring of varieties, equipped with the symmetric-power operations $\operatorname{Sym}^m$ and the involution…

Let $Q$ be a quiver, let $Q_\mathbb{Z}$ be its integral level quiver, and let $\mu(\mathbf{e})=\Theta(\mathbf{e})/\dim \mathbf{e}$ denote the slope of a dimension vector…

For $e\in \mathbb{Z}^n$ and $f\in \mathbb{Z}^m$, let … Here $P_s(t)$ is a polynomial, and $N_s$ is a sequence of integers with $N_s\to\infty$. Stabilization conjecture. The polynom…

Motivic PT generating-series conjecture. The generating series of the fiber-class invariants is given by the displayed infinite-product and plethystic-exponential formula.

Let $X$ be a smooth projective threefold over $\mathbb{R}$, equipped with an isomorphism … for some line bundle $L$ on $X$. For $n\geq 0$, let $\tilde{I}_n$ be the quadratic degree…

Refined Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

Let $k=\mathbb C$, and for every prime $p\geq 3$ let $G_p$ be the Heisenberg group of order $p^3$. Write $\{B_{\mathbb C}G_p\}$ for the motivic class of its classifying stack and…

Let a tropical curve be fixed, and consider the non-Archimedean semi-algebraic subset of the universal family of compactified Jacobians over the moduli space of stable curves of fi…

Let $G$ be a complex algebraic group, and let $\mathrm{K}\operatorname{M}_{G}$ denote the ambient module of motivic data associated with $G$. Define … the submodule generated by th…

Let $\Gamma$ be a tropical curve of genus $g-\delta$ and degree $\Delta$ through the points of $\operatorname{trop}(S)$. Let $|L|_{\Gamma}$ be the set of $K$-points in $|L|$ parame…

Let ${\mathscr{C}}$ be a suitable family of curves, and let $N_i({\mathscr{C}})$ denote the refinement of its motivic invariant obtained from the $\chi_{-y}$-genus. For a proper in…

Fix a quiver $Q$, a potential $W$, a stability condition $\zeta$, and a slope $\mu$. Let $\Lambda_{\mu}^{\zeta}$ be the set of dimension vectors of slope $\mu$, let…

Mozgovoy's conjecture. The invariants $\Om_D(r,d)$ are independent of $d$, without requiring $r$ and $d$ to be coprime, and are given by the formula proposed in the cited work. Thi…

Let $X$ be a smooth projective curve, let $K$ be its canonical divisor, and let $\cM_K(r,d)$ be the moduli stack of semistable $K$-twisted Higgs bundles over $X$ of rank $r$ and de…

Let $S$ be a K3 surface and let $\beta\in\operatorname{Pic}(S)$ be a positive class. Assuming Conjecture A, write $\mathsf{H}_{n,\beta}(S)$ for the virtual Poincaré polynomial inde…

Let $S$ be a K3 surface, let $\beta\in \operatorname{Pic}(S)$ be a positive class with divisibility $m_\beta$, and let $\mathsf{H}^\epsilon_{n,\beta}(S)\in \mathbb{Q}[u]$ denote th…

Let $X$ be a Calabi–Yau threefold, and let refined stable-pair invariants be the invariants constructed from the refined stable-pair theory described in the paper. Motivic stable-p…

Consider the configuration-space families discussed in the paper, with a limiting motive and corresponding limiting Poincaré series. Periodicity conjecture for limiting Poincaré se…

The $c_2$ residue conjecture. If $I_{G_1}=I_{G_2}$ for two primitive log-divergent graphs $G_1$ and $G_2$, then