14 problems

Whitney's fibering conjecture. Every analytic subvariety $V\subset U$ has a stratification such that each point $p_0\in V$ has a neighborhood $U_0$ with a semi-analytic fibration.

An analytic vector scheme is defined locally as the kernel of a map of finite-rank vector bundles. Conjecture on analytic vector schemes. This definition is equivalent to the notio…

Let $p_{cr}\ge 2$ and let $\Gamma$ be a compact analytic hypersurface. Let $M_\Gamma$ denote its associated maximal operator, let $\mu$ be the measure on $\Gamma$, and let…

Let $X$ be a pure $d$-dimensional $\mathbb{K}$-analytic set, where $\mathbb{K}$ is $\mathbb{C}$ or $\mathbb{R}$, and let $\mathcal{N}(X)$ denote its Nash transformation. A Nash–Sem…

Let $k$ be a complete non-Archimedean valued field, and let $f:X\rightarrow S$ be a quasi-finite morphism of quasi-compacted, separated $k$-analytic spaces. We seek a factorization…

Let $d\geq 2$, let $V$ be a periodic potential, and let $F_{\lambda}(V)$ denote the Fermi variety at energy $\lambda$. An analytic set is irreducible if it cannot be represented as…

Let $V$ be a periodic potential and let $B(V)$ denote its Bloch variety. An analytic set is irreducible if it cannot be represented as the union of two nonempty proper analytic sub…

Nonvanishing determinant conjecture. For every ${\mathcal I}\subseteq\{1,\dots,n\}$, the analytic function $\ell_{\mathcal I}({\bm \theta})$ is not identically zero.

Let $K$ be a Henselian valued field of characteristic zero with separated analytic structure, and let $A_{m,n}(K)$ denote the ring of separated analytic functions in $m$ variables…

Grothendieck's conjecture. The corresponding morphism of formal completions is again injective.

Generalized completion conjecture. Theorem should hold for $X$ over $k$ when the ideal $(f_1,\ldots,f_n)$ is replaced by $(t,f_1,\ldots,f_n)$.

Let $\mathbb{K}\{x\}$ be a convergent power series ring, let $I=(f_1,\dots,f_s)$ be an ideal, and for $\mu\in\mathbb{N}$ let $I_\mu$ be the ideal generated by the $\mu$-jets…

Let $(X,\sigma)$ be a quantized analytic cycle in a complex manifold $Y$. For any integer $p$ between $0$ and $r-1$, let $\lambda_p$ and $\mu_p$ be extension classes in … For integ…

Let $V$ be an analytic variety in $\mathbb C^n$ and let $X$ be a stratum of an analytic stratification of $V$. For every point $p\in X$, there should exist a neighborhood $U$ of…