5 problems

Let $K\{\mathbf{X};P\}$ be the analytic power-series ring over the valued field $K$, with $P$ the parameter set of valuation vectors, and let $I\subset K\{\mathbf{X};P\}$ be an ide…

Let ${\mathbf{k}}$ be an algebraically closed field of characteristic $0$, let $X=\mathcal{M}(A)$ be the affinoid space under consideration, and let $f:X\to X$ be an automorphism w…

Let $A,A_1,A_2,A_{12}$ be noetherian $\mathbb{E}_1$-Banach rings in the indicated simplicial ind-Banach categories. Suppose that their homotopy rings fit into a short strictly exac…

Tropicalization surjectivity conjecture. The map $\operatorname{Trop}$ is surjective. Moreover, if $g=\lvert\bm{\mu}\rvert$, then it is a bijection.

Let $k$ be a valued field, let $\mathfrak X$ be locally of the form $\operatorname{Spf}(\mathcal A^\circ)$ for $k$-affinoid algebras $\mathcal A$, so that $\mathfrak X$ is of finit…