36 problems

Nash's conjecture. The number of irreducible components of $X_{\infty}^{\operatorname{Sing}(X)}$ is at most the number of essential irreducible components of the exceptional locus…

The $(A_1,A_4)$ Gröbner-basis conjecture. The monomials in $\mathcal{G}(\ell,m)$ constitute a Gröbner basis for $\langle J(x)^2\rangle$, with the recursive description of…

The $(A_1,D_{2n+1})$ Zhu-algebra conjecture. The Zhu algebra of $\mathfrak{su}(2)_{-\frac{4n}{2n+1}}$ is isomorphic to $R$:

Let $f:\mathbb{C}^n\to\mathbb{C}$ be an algebraic map with an isolated singularity at the origin. Let $\mathcal{X}_m$ be the arc space of $f^{-1}(0)$, namely the space of maps from…

Nash's surjectivity conjecture. This map should be surjective.

Let $r\in\mathbf{Z}_{\geq 2}$, let $\mathcal{S}$ be the ring in which the ideal $\mathfrak{a}$ is defined, and let $J_r$ be the ideal generated by the monomials described above: th…

The Nash map associates to each irreducible component of the space of arcs of a singularity a divisorial valuation corresponding to an exceptional divisor in the minimal resolution…

Let $R$ be the coordinate ring of an affine scheme $X=\operatorname{Spec}(R)$, and let $J_\infty(R)$ be its arc space. Let $\operatorname{gr}(J_\infty(R))$ be the associated graded…

Let $\mathcal{T}$ be a four-dimensional $\mathcal{N}=2$ superconformal field theory, with Higgs branch viewed as an affine variety. For an affine scheme $X$, let…

στ-pizza completeness conjecture. Two normal pairs of Hölder triangles with the same στ-pizzas are outer Lipschitz equivalent.

Motivic jet-space integral conjecture. The identity

Let $I\subset k[\mathbf{x}]$ be a zero-dimensional ideal, where $\mathbf{x}=(x_1,\ldots,x_n)$, and let $h$ be a positive integer. For a positive integer $H$, consider the chain … a…

Isomorphism conjecture. The induced map $\hat{\psi}$ is an isomorphism.

Let $J^N$ be the finite arc algebra and let $\mathfrak{i}_k$ and $\mathfrak{j}_k$ be the support ideals generated respectively by the initial and terminal arc coefficients as defin…

Let $J[0^N,(1)^{N'}]$ and $A[0^N,(1)^{N'}]$ be the finite arc algebras, let $\mathfrak{a}$ be the local-cohomology support ideal, and let $\mathsf{th}$ denote the homomorphisms ind…

Let $J[\delta,(1)^N]$ and $A[\delta,(1)^N]$ be the finite arc algebras introduced in the source. Finite arc-algebra singularity conjecture. The algebra $J[\delta,(1)^N]$ is Cohen–M…

Let $\mathcal{C}$ be the cone in the source, let $\mathpzc{Arc}_q^N$ be the mapping space associated with the deformation $\mathbb{C}[z,q]/(z^{N+1}-q^{N+1})$, and set…

Threshold conjecture. One has

Intersection conjecture. For some $q$, the intersection

Nash's conjecture. Neither $\overline{N}_E$ nor $\overline{N}_F$ is included in the other:

Let $C$ be a connected curve over $k$ with only one singular point $p$, let $f:\bar C\to C$ be its normalization, and let $\bar\zeta_{C,p}(t)$ be the reduced auto-Igusa zeta series…

Let $C$ be a connected reduced curve over $k$ with exactly one singular point $p$, and let $\bar C\to C$ be its normalization of degree $e$. Let $\mathfrak{l}_{P_i(n)}$ denote the…

Let $C$ be the cuspidal cubic $y^2=x^3$, let $O$ be its origin, and let $\mathcal{A}_n(C,O)$ be its auto-arc space. Write $\mathcal{A}_n(C,O)^{\operatorname{red}}$ for its reductio…

Let $X$ be a variety over $k$, let $p$ be a point of $X$, and let $\mathcal{A}_n(X,p)$ be its auto-arc space at level $n$. Let $\kappa(p)$ be the residue field at $p$, and write…

Let $X$ be an algebraic variety over $k$, let $p$ be a point of $X$, and let $\mathcal{A}_n(X,p)$ be the auto-arc space at level $n$. Let $\mathcal{A}_n(X,p)^{\operatorname{red}}$…