35 problems

van Straten–Warmt conjecture. The odd Betti numbers of $J_{\mathcal{R}}$ vanish; equivalently,

Ribbon-genus conjecture. The ribbon genus of $\operatorname{Bd} F_0$ is at least

Morsification conjecture for topological exceptional collections. The ordered collection of vanishing cycles obtained from a topological exceptional collection of geometric vanishi…

A real plane curve singularity may have pairs of complex conjugate local branches, and its divide may consequently contain immersed circles. Lissajous-block conjecture. If the real…

The differential-value conjecture. For any element $\lambda\in\Lambda_C\setminus\Gamma_C$, the rational number $-\lambda/nm$ is a root of the Bernstein–Sato polynomial of $C$.

Let $\Gamma$ be the value semigroup of a plane curve singularity, let $\mathcal{\Delta}=\{\Delta_0\subset\cdots\subset\Delta_\ell\}$ be a standard increasing sequence of $\Gamma$-m…

Let $\mathcal{H}^{mot}(q,t,a)$ be the motivic superpolynomial of a plane curve singularity, and let $L(q,t,a)$ denote the corresponding generalized Galkin–Stöhr $L$-function. Secon…

Let $f\in\mathbb{K}[[x,y]]$ be a reduced power series, and let $\mathcal{T}$ be a minimal tree of $f$. Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$, let…

Let $R$ be a plane curve germ over $\mathbb{C}$, and let $\widehat{\mathit{NZ}}_R(t)$ denote its completed numerator series. Modularity and topological invariance conjecture. There…

Quasi-polynomial determinant conjecture. The determinant $\det S_k$ is a quasi-polynomial in $k$ of period $N$.

A real morsification is a real morsification of a real isolated plane curve singularity, and each such morsification has an associated quiver. Fomin–Pylyavskyy–Shustin conjecture.…

Topological monodromy conjecture. If $\theta$ is a pole of $Z_f^{\mathrm{top}}(s)$, then $\exp(2\pi i\theta)$ is an eigenvalue of the monodromy $M_x^{(q)}$ for some $q\in\mathbb N$…

The paper studies plane curve singularities over an algebraically closed field and distinguishes positive characteristics according to whether they are good or bad for the deformat…

Restricted Yano conjecture. The $b$-exponents $\{\tilde{\beta}_1,\ldots,\tilde{\beta}_\mu\}$ of a generic element of $\Sigma_\mu$ satisfy

An algebraic divide is a divide arising from a real morsification of a plane curve singularity. A divide is malleable when it is YB-equivalent to a scannable divide, where YB-equiv…

Let $D$ and $D'$ be link-equivalent scannable algebraic divides of the same minimal index. Let $\beta(D)$ and $\beta(D')$ be their associated positive braids; two positive braids a…

Let $(C,z)$ and $(C',z')$ be real isolated plane curve singularities, and choose real morsifications of them. Each morsification has an associated quiver; two quivers are mutation…

Admissible-flag conjecture. The only admissible flags for the family $\mathbf{C}[t^4,t^6+t^v]$ are those from Table 4–6.

Cell-type conjecture. Table 4–6 contains all possible types of cells, for any flags, that can occur for the whole series $\langle4,6,6+v\rangle$.

Singular superpolynomial conjecture. For all $i,d\geq0$, $H^{\!BM}_{2i+1}(\mathcal{J}^{\ell}_{\mathcal{R},{\mathsf \varrho}}[d])$ vanishes, and

Existence conjecture. Every real plane curve singularity possesses a real morsification.

Main conjecture. For all $i,d\geq0$, the relative homology $H_{2i+1}(\overline{J}^m_{\mathcal{R}}[d],\overline{J}^m_{\mathcal{R}}[d+1])$ vanishes, and

ORS conjecture. The geometry of nested Hilbert schemes is related to the unreduced stable Khovanov–Rozansky polynomials.

Let $\mathcal{C}_{\vec{\mathsf r},\vec{\mathsf s}}$ be a unibranch plane curve singularity, let $J(\mathcal{C}_{\vec{\mathsf r},\vec{\mathsf s}})$ be its Jacobian factor, and write…

Let $C$ be a locally planar rational curve with a unique singularity, and let $\Lambda$ be a rainbow closure of a positive braid such that the topological knot underlying $\Lambda$…