22 problems

Let $G$ be a connected reductive complex algebraic group, let $K$ be a maximal compact subgroup of $G$, and let $X$ be a spherical homogeneous algebraic variety. Let…

General semisimple-group conjecture. Theorems on the contractions and product structure—and consequently the orientability theorem and the relative-homology corollary—hold for any…

Let $X$ be a stratified space with stratification ${\cal S}$, let ${\cal P}(X,{\cal S})$ denote MacPherson's category of perverse sheaves, and let $D^b_{\cal S}(X)$ denote the boun…

Stratified latent-space conjecture. Distinct strata in the latent space correspond to different state-action trajectories, with increases in local dimension occurring when the agen…

Stratified obstruction theory. For each $\tau\in(0,2]$, the space $\mathcal{U}_\tau$ contains countably infinite isotopy classes. As $\tau\to0$, geometric obstructions, including b…

Let $\mathcal{S}\mathrm{trat}$ denote a homotopy theory of topological stratified spaces, and let $\iCat$ denote the homotopy theory of $\infty$-categories. The exit path construct…

Let $M$ be a Scott complex or quaternionic manifold with its natural stratification. Let $N$ be a stratified space and let $f:N\to M$ be a stratified homotopy equivalence. Stratifi…

Let $M^n$ be a manifold with a “nice” group action or a space with a “nice” stratification, and let $N^n$ be another manifold or space that is equivariantly or stratified homotopy…

Let $M$ be a smooth manifold with a proper action of a group $G$. For each $x\in M$, let $N_x(M,G\cdot x)$ be the normal space to the orbit through $x$, and define … Let $G_x$ be t…

Categorical compactness conjecture. If $(X,P)$ is of finite stratified type, then $(X,P)$ is also categorically compact.

Let $(X,\mathcal{C}^\infty)$ be a homologically regular reduced differentiable stratified space, possibly belonging to an allowable class such as locally semialgebraic differentiab…

A complex analytic variety is a space locally defined by complex analytic equations. A Whitney stratification is a decomposition into smooth strata satisfying Whitney's conditions;…

Let $f:X\to Y$ be a cylindrically stratified morphism of stratified topological spaces. A morphism is compactifiable if it factors as a stratified open embedding followed by a prop…

Sphericality conjecture. For any good stratification $A$, this pullback-limit adjunction is spherical and arises as the restriction of a spherical adjunction involving the sheaf in…

Let $X$ be a stratified space, let $R$ be a commutative ring, and let $\overline{p}$ and $\overline{q}$ be Goresky–MacPherson perversities. Intersection cohomology operations are n…

In Hamiltonian lattice gauge theory, let the configuration space be decomposed into gauge orbits with a stratification into orbit-type strata, and let the quantum Hamiltonian have…

Stickiness conjecture. This phenomenon is related to the phenomenon of stickiness of the Fréchet mean in stratified spaces.

Let $z_n$ denote the logarithmic fluctuation associated with the empirical Fréchet mean on a Riemannian manifold, and let the first moment (bias) be…

Let $(X^n,g)$ be an admissible stratified space: if it has a codimension-two stratum, its link has diameter less than $pi$, and its iterated edge metric satisfies…

Exponential-law conjecture. The functor $\underline{\pi_1}_*$ is an equivalence under rather weak conditions on $X$ and $Y$.

Let $(M,g)$ be a smoothly stratified Riemannian pseudomanifold with only isolated conic singularities. Let $Y(M,[g])$ denote the Yamabe constant of the conformal class $[g]$, and l…

Tameness-free exit-path conjecture. The latter tameness condition can be removed.