35 problems

Let $\mathcal{M}$ be the subcategory of $C^n$-manifolds, where $n\in\mathbb{N}$ or $n=\infty$, consisting of all manifolds but only diffuse $C^n$-functions. Let the elementary fini…

Let $M$ be a closed manifold with zero Euler characteristic, and let $\alpha$ be a non-closed differential $1$-form on $M$. Non-vanishing perturbation conjecture. There exists a smo…

Conjecture on negative transverse curvature. There is a Riemannian metric $g$ on $M$ such that the function $\chi_{g,\bot H}$ is everywhere negative.

Let $M$ be a compact manifold of dimension $m$, let $P:M\rightarrow\mathbb{R}^d$ be smooth, and let $\mathfrak{F}$ be the quasi-stratification of $M$ by the level sets…

Conformal concordance conjecture. If $C_0$ and $C_1$ are conformally concordant, then they are isotopic in ${\cal C}^+(M)$.

A homotopy 4-sphere is a smooth 4-manifold homotopy equivalent to the 4-sphere $S^4$. Smooth 4-dimensional Poincaré conjecture. Every homotopy 4-sphere is diffeomorphic to $S^4$. T…

Yau's conjecture. For $n\geq 3$, any closed $2n$-manifold admitting an almost complex structure will also admit a complex structure.

Let $\mathsf{W}_{\mathrm{CC}}$ be the space of continuous-contingent microbundles, let $u\in\mathsf{W}_{\mathrm{CC}}$, and let $\mathrm{D}(u)$ denote its domain. A path $\omega$ de…

Let $M$ be a closed manifold, and let $\mathrm{Riem}^+(M)$ denote the space of positive scalar curvature metrics on $M$. Two metrics in this space are concordant if there is a posi…

Let $M$ be a compact differentiable manifold, and let $G$ be a finite subgroup of the diffeomorphism group of $M$. Ghys's conjecture. There is a constant $I$ such that $G$ has a ni…

Generalized Smale Conjecture. This inclusion is a homotopy equivalence. The conjecture was verified by Kleiner and Bamler, giving a complete description of the relevant homotopy ty…

Let $M$ and $N$ be closed $d$-manifolds with $d \geq 5$ and the same normal $2$-type. Assume that both $M$ and $N$ admit metrics of positive scalar curvature. Write…

Let $(V,\partial V)$ be a manifold of dimension at least $3$. Let $X\to B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is any manifold with a point-s…

A complete intersection $X_n(\underline d)\subseteq\mathbb{C}P^{n+k}$ is the transverse intersection of $k$ complex hypersurfaces of multidegree $\underline d=(d_1,\ldots,d_k)$. Le…

Let $V:\mathbb R^3\to S^2$ be a unit vector field defining a fibration of $\mathbb R^3$ by oriented lines. Put $U=V(\mathbb R^3)$, and for $u\in U$ let $u^\perp$ be the plane throu…

Great-circle conjecture. If there exist multiple $u\in U$ for which $S_u$ is noncompact, then the set of such $u$ is contained in some great circle in $S^2$.

Let $M$ and $N$ be $4$-dimensional homeomorphic closed and connected manifolds, and suppose that each admits a special generic map into some Euclidean space of dimension smaller th…

Rank-one approximation conjecture. The mapping $f$ can be uniformly approximated, at least locally, by mappings $g\in C^\infty(\mathbb R^n,\mathbb R^k)$ satisfying…

Gałęski's local approximation conjecture. Every such mapping admits, around each point, a uniformly convergent smooth approximation sequence preserving the derivative-rank bound.

Lakzian's conjecture. The space $M$ is diffeomorphic to Euclidean space $\mathbb{R}^{n}$.

The Prem Conjecture. Smoothly $k$-realizable general-position smooth maps $N^n\to\mathbb{R}^m$ are smooth $k$-prems, at least in the metastable range

Metric-invariance conjecture. The homology groups of these differential complexes depend only on the connected component of the space of traversally generic metrics on $M$ containi…

Codimension-1 Grassmannian embedding conjecture. The only codimension-$1$ smooth embedding

Let $l\geq k>0$, fix an o-minimal structure, and let a derived $C^k$-manifold be a manifold in the corresponding derived differentiable setting. Whitney smoothing conjecture. Every…

Rigidity conjecture. $M^n$ is equivariantly diffeomorphic to an effective, linear action of $T^k$ on a manifold of one of the following forms: