41 problems

Let $x\thicksim X$ be a symplectic singularity. Let $v_X$ be the valuation associated with $x$ in Theorem AGlc, and let $r(\xi)$ denote the rational rank associated with $\xi$ in T…

Let $\Delta_{\ell-1}$ and $\Delta_\ell$ be the value groups appearing in the tight extensions, let $\hat{\mathcal P}”_{\beta\ell}$ and $\hat{\mathcal P}'_{\beta\ell}$ be the corres…

SNC index specialization conjecture. Under these assumptions,

Let $K=K\{{\mathbf{f}}\}$ be a Poisson field, let ${^1\Gamma}_0(K)$ be the $1$-valuation gamma-cap, and let $\operatorname{fht}(K)$ denote the flag height. Gamma-cap conjecture. If…

Liu–Xu's triangulation conjecture. There is a rational triangulation of $\mathcal{D}^{KV}(X,D+\Delta)$ such that, in the interior of $\mathcal{D}^{KV}(X,D+\Delta)$, the triangulati…

Let $n$ be a positive integer, let $\mathcal{P}^n$ denote the family of polytopes in $\mathbb{R}^n$, and let $\mathcal{M}(\mathbb{R}^n)$ denote the space of real-valued functions o…

Let $L_0$ be a field and let $L_1$ be a nontrivial finitely generated extension of $L_0$. Suppose there exists a valuation $v$ on $L_1$ with residue field $L_0$. Weak resolution of…

Let $G/H$ be a spherical homogeneous space, let $K$ be a maximal compact subgroup of $G$, and let $\mathcal{V}_{G/H}$ be the valuation cone. For each face $\sigma$ of…

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

Measure-classification conjecture. Then $\mu$ is a constant multiple of Lebesgue measure.

Log uniformization conjecture. Every valuation on a quasi-excellent integral scheme $X$ is log uniformizable.

Let $n$ be a positive integer, let $0\leq r\leq\lfloor\frac n2\rfloor$, and let $\mathcal K(\mathbb R^n)$ denote the space of convex bodies in $\mathbb R^n$. For $i=0,\dots,n-2r$,…

Recursive uniformizer and stability conjecture. For each prime $p\geq 3$ and integer $n\geq 1$, there exists a series of polynomials $\{\mathcal R_p^{m,n}(T)\}_{m\geq 2}$ such that…

Let $b[?]$ be a valuation and $Q$ an abstract key polynomial (ABKP) for $b[?]$. For an associated diskoid $b?)$, let $b[?]_{b?)}$ denote the induced function on $b[?][X]$. An RT ex…

Let $b[?]$ be a valuation on $K[X]$ and let $Qb[?]K[X]$ be an abstract key polynomial for $b[?]$, with $b?b[?]b[?]b?$. A diskoid is a subset $b?b[?]b[?]$ whose polynomial-value min…

Finite-generation conjecture. The graded ring ${\rm gr}_vR$ is finitely generated.

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Let $v:2^{\mathcal G}\to\mathbb R$ be a valuation, and define the merging operation (or convolution) of valuations $v_1,v_2$ by … An $M^\natural$-concave valuation is a valuation i…

Let $v:2^{\mathcal G}\to\mathbb R$ be a valuation. An endowment operation with respect to $T\subseteq\mathcal G$ produces the valuation $v'(X)=v(X\cup T)-v(T)$, and a Rado minor va…

3-adic valuation pattern conjecture. There exists a 3-adic integer $\alpha$ such that, for odd $n\geq 3$, the quantity $\xi_3(L_n-1)$ depends only on $\xi_3(n-\alpha)$ and the last…

Postnikov's 5-adic valuation conjecture. For $n\geq 4$,

Let $(K,\nu)\subset (L,\omega)$ be in situation, with valuation rings $\mathcal{O}_\nu$ and $\mathcal{O}_\omega$. Assume in addition that the field extension $L|K$ is separable. Se…

Let $K$ be a field, let $R\subseteq K$ be a good Bezout domain, and let $I_M$ denote the set of $M$-infinitesimals. The $M$-infinitesimals $I_M$ are an ideal in $R$. This would pro…

Let $M$ be a Riemannian manifold. A valuation $\mu$ is called angular when … where $\mathcal{A}(M)$ is the space of angular curvature measures and $\cdot$ denotes the Alesker produ…