19 problems

Fixed-point formula conjecture. The motivic generating series satisfies

Let $G$ be a connected linear algebraic group over a field $k$, and let $M_g$ be a smooth compact surface of genus $g$. Write $\operatorname{Rep}_G(M_g)$ for the $G$-representation…

For a positive integer $n$, let $C_n(\operatorname{GL}_r(k))$ denote the variety of commuting $n$-tuples in $\operatorname{GL}_r(k)$. Motivic stability conjecture. The limit … exis…

Mozgovoy's motivic ADHM conjecture. For every $r\geq 1$, $H_r(t)$ is a polynomial in $t$ and

For each positive integer $d$, define … Let $\zeta_{{\mathbb P}^{d-1}}(t)$ denote the zeta function of projective $(d-1)$-space. Omega-class generating-function conjecture. Fix…

For positive integers $n$ and $d$, let $\Omega^n_d$ be the effective classes defined by … and let $\chi$ denote the Euler characteristic. Polynomial-error conjecture. For every…

Let $\mathsf P_d(t)$ denote the polynomial in the motivic formula for Hilbert schemes of $d$ points, and suppose that $\mathsf P_e(t)$ is known for every integer $e$ with…

For each positive integer $d$, let $\mathsf P_d(t)$ be the polynomial appearing in the motivic formula for Hilbert schemes of $d$ points on smooth varieties. Exact-degree conjectur…

For $m\geq 1$ and $d\geq 0$, let $R=R^{(2,2m)}$. Let $\mathit{NZ}_{R^d}(t)$ be the motivic numerator and $\widehat{\mathit{NZ}}_R(t)$ its completed version. Positivity conjecture.…

Local functional equation conjecture. As rational functions in $t$,

Global functional equation conjecture. As rational functions in $t$,

Let $\mathcal{E}$ be an $\alpha$-twisted locally free sheaf on $\operatorname{Spec}K$ of rank $n$, and let $\mathbb{P}\mathcal{E}$ denote its associated twisted projective space. T…

Let $Q$ be a quiver with an automorphism $\ta$, let $\bv\in\bN^{(Q_0)}$ be a dimension vector, and let $\cR^\ta(Q,\bv)$ denote the space of pairs $(M,\vi)$ with $M\in\cR(Q,\bv)$ an…

Let $M$ be a smooth projective variety endowed with a faithful action of a finite group $G$, and let $X:=M/G$ be its quotient. Assume that $X$ is Gorenstein, and let $Y\to X$ be an…

Kernel generating-series conjecture. The following identities are expected:

Let $k$ be a field, let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$, and let $\mathbb{L}=[\mathbb{A}^1_k]$ be the Lefschetz class. Cancellation conje…

Let $k$ be a field, and let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$. Liu–Sebag's cut-and-paste conjecture. If varieties $X$ and $Y$ over $k$ sati…

Let $k$ be a field, let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$, and let $\mathbb{L}=[\mathbb{A}^1_k]$. Let $\hat K$ be the completion of…

Let $a,b$ be positive integers, let $({\mathbb C}^2)^{[n]}_{a,b}$ denote the Hilbert scheme of $n$ points on the plane fixed by the quasihomogeneous action with weights $(a,b)$, le…