6 problems

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…

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…

Let an affine singularity be an affine algebraic variety with singularities. An embedding means placing it in a larger ambient space, and a toric map is a morphism arising from tor…

Analytic local uniformization conjecture. Every point $x\in X^{\operatorname{ad}}$ is uniformizable.

Local uniformization conjecture. For every valuation $\nu$ of $K/k$, there exists a projective model $M$ of $K/k$ on which the foliation $\mathcal{F}$ is log-final at the center of…

Tight extension conjecture. The valuation $nu$ admits at least one tight extension $hat\nu_-$. It can be chosen so that, for rings $R'$ sufficiently far in $cal T$,