12 problems

Sard conjecture. For every $x\in M$ and every integer $r\in[m,n-1]$, the set $\operatorname{Abn}^{r}_{\Delta}(x)$ has zero Lebesgue measure in $M$.

Let $R$ be the coordinate ring of an affine reduced singular curve over $\mathbb{F}_q$, let $\widetilde R$ be its normalization, and write $\widehat{Z}_R(t)$ for the Cohen–Lenstra…

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.…

For $m\geq 1$, let $R^{(2,2m)}$ denote the even plane curve singularity and let $\widehat{\mathit{NZ}}_{R^{(2,2m)}}$ be its completed numerator series. Rogers–Ramanujan-type conjec…

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

Let $R$ be a planar singularity, let $d\geq 0$, and write … for its Serre invariant. Define the numerator part of the Quot zeta function by $\mathit{NZ}_{R^{\oplus d}}(t)$. Functio…

Let $C$ be a Gorenstein unicuspidal rational curve of genus $g$, and let $\gon_F(C)$ denote its covering gonality, namely the smallest degree of a map from $C$ to $\mathbb{P}^1$. C…

The stability-manifold conjecture. The action of $\mathrm{Auteq}(C)$ on $\mathrm{Stab}(C)$ preserves $\mathrm{Stab}^{dagger}(C)$, and $\mathrm{Stab}^{dagger}(C)$ is simply connecte…

Minimal rank Sard conjecture. For every $x\in M$, the set $\operatorname{Abn}^{m}_{\Delta}(x)$ has zero Lebesgue measure in $M$.

Kleiman–Piene equisingularity conjecture. Universal polynomials in the pushdowns of products of the relative Chern classes of the family should enumerate curves with any given equi…

Let $\delta$ be a nonnegative integer, and let the polynomials supplied by Göttsche's universal formula count $\delta$-nodal curves in $\mathbb{P}^2$ of degree $d$. A polynomial is…

Let $C$ be a rational planar projective reduced curve of arithmetic genus $\delta$, and let $C^{[n]}$ denote its Hilbert scheme of length-$n$ subschemes. Write…