82 problems

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…

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…

Montgomery–Yang problem. Every pseudofree $\mathbb{S}^1$-action on $\mathbb{S}^5$ has at most $3$ non-free orbits.

Let $f:M\rightarrow N$ be a homotopy equivalence between closed negatively curved manifolds. The unique harmonic map $h$ homotopic to $f$ is thereby defined. Lawson–Yau conjecture.…

Density conjecture. $\mathcal M^\ddagger(M)$ is dense in the space of metrics of the gradient type.

Friedman–Morgan conjecture. Two algebraic surfaces are diffeomorphic if and only if they are deformation equivalent.

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 $KT^7_{1,3,DIFF}$ denote the set of smooth embeddings $S^1\times S^3\to\mathbb{R}^7$ up to smooth isotopy. The smooth knotted-torus classification conjecture. The set…

Higher-regularity homogeneity conjecture. The analogue of the theorem holds in the $C^r$ category for every $r\ge2$ and in the analytic category.

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 smooth compact simply connected spin $4$-manifold, with Euler characteristic $\chi(M)$ and signature $\tau(M)$. The $11/8$ conjecture asserts that … Equivalently, ever…

For every integer $m>10$, there is a harmonic cellular map $h:M_1\rightarrow M_2$ between a pair of closed negatively curved $m$-dimensional Riemannian manifolds that is not a diff…

Let $M$ and $N$ be closed negatively curved Riemannian manifolds, and let $h:M\to N$ be a harmonic map that is not a diffeomorphism but is the limit of a 1-parameter family of diff…

Let $P$ and $Q$ be manifolds, let $f\colon P\to Q$ be a general-position mapping, and let $I_k=(p-q+1,1,\ldots,1,0)$ have $k$ non-zero entries, with $p-q$ odd and $k\geq 4$. A Mori…

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 ${\cal C}^+(M)$.

Zero-or-undefined index conjecture. Any pseudo vector field must have either index equal to zero or an undefined index.

Nonexistence conjecture. There is no $\mathbf{E}_7$-structure on $S^{133}$.

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

Conjecture on corank-2 fat distributions. The $h$-principle fails for fat distributions of corank $k>1$, at least when $k=2$.

Unique-smooth-structure conjecture. These are the only values of $k$ for which $S^{k+1}$ has a unique smooth structure.

An Engel structure is a maximally non-integrable rank-2 distribution $\mathcal{D} \subseteq TN$ on a $4$-manifold $N$. Such structures have no local invariants beyond their local n…

Topping's conjecture. The manifold $M$ should be diffeomorphic to $\mathbb R^n$.

Hirsch conjecture. There is an immersion of $M$ in $\mathbb{R}^n$ which is regularly homotopic to an embedding. This is a reformulation of the Hirsch conjecture, which concerns emb…