72 problems

Let $(R, \mathfrak{m}_R, k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero, and let $\Omega_R$ denote its universally finite module of…

Let $J_{\bullet}$ be a graded sequence of ideals on $X$, let $\mathfrak{q}$ be a nonzero ideal on $X$, and suppose that the asymptotic jumping number…

Let $V$ be an $n$-dimensional Euclidean vector space and let $G$ be a compact group acting transitively on the unit sphere of $V$. Write $Val_i^G(V)$ for the degree-$i$ component o…

Let $(K,v)$ be a Henselian valued field, and let $L/K$ be a finite simple field extension such that $(L/K,v)$ is defectless. Let $r$ and $s$ be the minimal numbers of generators of…

McMullen's conjecture. Mixed volumes span a dense subspace of $\operatorname{Val}(\mathbb{R}^n)$.

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…

Let $M$ be a Riemannian manifold. The space of angular curvature measures is denoted by $\mathcal{A}(M)$, and the Lipschitz–Killing algebra by $\mathcal{L}\mathcal{K}(M)$. Here a v…

Li's normalized-volume conjecture for the exceptional weighted valuation. For all other cases of $(n,k)$, ${\widehat{\rm vol}}$ should be minimized at the valuation associated to t…

Li's normalized-volume minimization conjecture. The normalized volume should have a global minimizer on ${\rm Val}_{Y,o}$ for every ${\mathbb Q}$-Gorenstein klt singularity $(Y,o)$…

Let $\mathcal{N}$ be the category of all noetherian local domains. A valuation $\nu$ centered on a local integral domain $R$ admits local uniformization if there exists a local rin…

Scalewise birational extension conjecture. There exists an ideal $H\subset\hat R$ with $H\cap R=(0)$ and a valuation $\hat\nu_-$ centered at $\hat R/H$ such that

SNC index specialization conjecture. Under these assumptions,

Let $V$ be an $n$-dimensional Euclidean vector space, and let $V_1\in Val_1^\infty(V)$ denote the first intrinsic volume. For each integer $i$ with $0\leq i<n/2$, consider multipli…

Let $R\rightarrow S$ be a birational extension of regular local rings of dimension $n\geq 3$, and let $V$ be a valuation ring of the quotient field of $S$ that dominates $R$. A mon…

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…

Let $R$ be an excellent equicharacteristic Noetherian local ring, let $\widehat{R}$ be its completion, and let $\nu$ be a rational valuation of $R$. Teissier's extension conjecture…

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 $\mathbb{A}$ be the underlying measurable space, and let $\mu$ be an s-finite valuation on $\mathbb{A}$. A Poisson point process with expectation measure $\mu$, denoted…

Matyas's conjecture. If $\varrho$ is a homogeneous polynomial of degree at least $4$ satisfying the displayed functional equation, then it satisfies the displayed identity. The que…

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…

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

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

Knaf's conjecture. If

Let $R$ be a complete equicharacteristic local domain with algebraically closed residue field, and let $u$ be a rational valuation centered in $R$ of rational rank $r<dim R$. Let…