116 problems

Wahl's conjecture. Every rational homology disk singularity is weighted homogeneous.

Let $f\in\mathbb{C}[[x,y,z]]$ be an isolated hypersurface singularity. Let $\mu$ be its Milnor number and $p_g$ its geometric genus. Durfee's conjecture. … with equality only when…

Rational surface singularities splice-quotient conjecture. Every rational surface singularity is a splice-quotient singularity.

Casson invariant conjecture. The Casson invariant of $a3$ equals one-eighth the signature of the Milnor fiber.

Frobenius liftability conjecture. Every cusp singularity is $F$-liftable.

Let $(X,0)$ be a Gorenstein normal surface singularity with link $M$, and suppose that $M$ is a rational homology sphere. A property is topological if it can be determined from the…

Casson invariant conjecture. The Casson invariant of $\partial(X,o)$ is equal to one-eighth of the signature of the Milnor fiber of $(X,o)$.

Birbrair–Gabrielov conjecture. If $(X_0,0)$ and $(X_1,0)$ are ambient topologically equivalent and outer bi-Lipschitz equivalent, then $X_0$ and $X_1$ are ambient bi-Lipschitz equi…

Let $(X,0)$ be a rational surface singularity, and consider its resolution. Assume that every exceptional curve in the resolution has self-intersection at most $-5$, and let the ba…

Let $(X,D,x)$ be a germ of log canonical surface singularity with $\mathbb{Q}$-boundary. Borbon–Spotti conjecture. … The conjecture relates Langer's local orbifold Euler number to…

Let $(X,0)$ be a cyclic quotient surface singularity of type $\frac{1}{n}(1,q)$ with $(n,q)=1$, let $L(n,q)$ be its link, and let $\mathcal{Z}_e\left(\frac{n}{n-q}\right)$ be Lisca…

Yau's positivity conjecture. This invariant is positive for every normal surface singularity.

Let $(X,E)\rightarrow (V,0)$ be the minimal good resolution (MGR) of a complex normal surface singularity, and define $S_X=(\Omega^1_X(\operatorname{log} E))^*$. Assume that…

Némethi–Nicolaescu conjecture. For certain singularities,

A rational double point is a surface singularity of type $A_{n-1}$ or $D_n$, and a partial resolution dominated by its minimal resolution has an associated maximal Cohen--Macaulay…

Fix a rational homology sphere link and consider the general' $\mathbb{Q}$-Gorenstein singularity with that link. Suppose that its fundamental cycle has arithmetic genus at least…

Let $(V,0)$ be a hypersurface surface singularity, and let its Milnor fibre carry its intersection form with signature defined in the usual way. Durfee's conjecture. The signature…

A superisolated surface singularity has an abstract topology, a characteristic polynomial of monodromy, and an embedded topology. Yau's conjecture. Abstract topology and the charac…

Let $(Y,0)$ be a normal surface singularity with rational homology sphere link, and let $tilde{Y}$ be its minimal good resolution. An end-curve function for an end-curve is a funct…

Let $p>q\geq 1$ be coprime integers, let $L(p,q)$ be the corresponding lens space with contact structure $\overline\xi_{\rm st}$, and let $\mathbf Z_{p,q}$ denote the set indexing…

Let $\Sigma$ be obtained by splicing homology spheres $\Sigma_1$ and $\Sigma_2$ along an edge of a splice diagram satisfying the semigroup condition. Let $F$, $F_1$, and $F_2$ be t…

Complete-intersection semigroup-condition conjecture. If $X$ is a complete intersection, then $\Delta$ satisfies the semigroup condition.

Let $(X,o)$ be a Gorenstein surface singularity with integral homology sphere link, and let its splice diagram encode the topology of the link. Gorenstein splice-type conjecture.…

Let $(X,o)$ be an isolated complete intersection surface singularity whose link $\Sigma$ is an integral homology $3$-sphere. The Casson Invariant Conjecture. The Casson invariant…

Let $Y$ be a log Enriques surface, let $Y^0$ be its smooth locus, and let $U$ be the universal cover of $Y^0$. A big open set means the complement of a discrete subset; a Du Val…