16 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.

Let a sub-Riemannian sphere be the metric sphere associated with a sub-Riemannian structure, and consider the general case with abnormal minimizers. The log-exp category is the fun…

Proper stratification conjecture. The preorder of is properly stratifying for $A_0(\vecn)$ in arbitrary characteristic.

Countable stratification conjecture. The category $\Cell{G}$ is countably stratified for every finite group $G$.

Let an atomic-strata sampling design use a simulated annealing method that selects atomic strata for exchange. Atomic strata near stratum boundaries are distinguished from more imp…

All-symmetries conjecture. For each $k\leq n-2$, there exists $\hat{\mathcal{S}}^k$ with $\dim\hat{\mathcal{S}}^k\leq k$ such that for each $x\in X\setminus\hat{\mathcal{S}}^k$, ev…

Stratified manifold conjecture. $\mathcal{S}^k$ is contained in a union of $k$-dimensional submanifolds. More precisely, there exist a countable collection of bi-Hölder maps…

Let $S_{n+1}$ be the symmetric group, let ${\mathbf W}_n$ be the set of words, and let $\operatorname{inv}(\sigma)$ denote the inversion number of $\sigma$. For a nonidentity permu…

Let $S_{n+1}$ be the symmetric group, let ${\mathbf W}_n$ be the set of words in the relevant alphabet, and let $\operatorname{mult}:S_{n+1}\to\mathbb{N}^n$ be given by … Extend th…

Rigidity Conjecture. The topological class is a finite-codimension manifold which is stratified by probabilistic rigidity classes.

Many-exponential transversality conjecture. Transversality remains true in the general case of many exponentials.

Let $\mathcal{C}$ be a compact family of closed cones. Suppose that at each point of a set $S$, each blow-up of $S$ is contained in some element of $\mathcal{C}$. Let $d$ be the la…

Let $C$ be a closed cone in $\mathbb{R}^n$. Its building dimension is … where $\operatorname{diff}(S):=\{x-y:x,y\in S\}$. White's building-dimension conjecture. The building dimens…

Let $W$ be a compact manifold of dimension $n\geq m$, let $u\in C^\infty(W,\mathbb{R}^m)$, and let $\theta$ be its Pareto critical set, with $\theta_S$ denoting the singular part o…

Let $(A,\Sigma)$ be a differentiable stratified space $b$-regularly embedded by $p$ into $\mathbb R^n$: for strata $Y<X$, whenever sequences in $X$ and $Y$ converge to a point of…

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…