22 problems

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 $f\colon M\to\mathbb{R}$ be a smooth function on a manifold $M$ without boundary, and let $x_0\in M$ be an isolated critical point. A cylindrical ball neighborhood is a cylindr…

Nash's rationality conjecture. There exists a rational variety $X$ whose real locus is diffeomorphic to $M$:

Let $M$ be a compact smooth $k$-manifold and let … be a smooth embedding. A tame $k$-tangle is the relevant tame embedded object in $lR^n$. Smooth embedded manifolds are genericall…

Let $M$ and $M'$ be smooth manifolds with smooth embeddings … Assume that these embeddings correspond to the same normalized tangle truss in $lR^n$. Framed stratified homeomorphism…

Let $M$ be a smooth compact manifold smoothly embedded in euclidean space. Framed stratifications are dense in smooth embeddings. Any smooth embedding of $M$ into euclidean space h…

Let $M$ and $M'$ be smooth compact manifolds smoothly embedded in euclidean space, and let the embeddings define flat framed stratifications. Framed homeomorphism implies diffeomor…

Let $M$ be a smooth manifold. A flat framed stratification is a stratification arising in flat framed euclidean space, and its associated truss is the corresponding combinatorial r…

Suppose $M$ is a locally symmetric space with natural Riemannian metric $\rho$ and Levi-Civita connection $abla$. Let $f$ be a self-diffeomorphism of $M$, and let $(g{1,b})$ and…

Let $X_4(\underline d)$ be a $4$-dimensional complete intersection, let $v_2(X_4(\underline d))$ and $v_4(X_4(\underline d))$ denote its Wu classes, regarded as elements of…

Wang–Xu conjecture. The dimensions without exotic spheres are exactly $1,2,3$, possibly $4$, and $5,6,12,56,61$.

Conjecture I. It is not possible to have a smooth faithful action of a genuine compact quantum group on $C(M)$.

Smooth manifold set cardinality conjecture.

Let $\Sigma$ be a closed connected smooth $n$-dimensional manifold. Nash's theorem gives a smooth embedding of $\Sigma$ into $\mathbb R^{2n+1}$ whose image is a connected component…

Let a Diophantine property be a property of vectors in an ambient space such as $\mathbb{R}^{n}$, and let $\mathcal{M}$ be a nondegenerate smooth submanifold of that space. Diophan…

A spacetime is modeled by a smooth four-manifold, and quantum gravity is obtained by combining this geometric model with quantum field theory and a quantization of general relativi…

Let $M$ be a compact connected smooth manifold, and let a genuine compact quantum group act smoothly and faithfully on $M$. Goswami–Joardar's rigidity conjecture. No such smooth fa…

Conjecture I. It is not possible to have smooth faithful actions of genuine compact quantum groups on $C(M)$ when $M$ is a compact connected smooth manifold.

Existence conjecture. Every closed smooth $n$-dimensional manifold admits a completely hyperbolic nondegenerate action of $\mathbb{R}^n$.

Let $M$ be a compact connected smooth manifold, and let a compact quantum group act smoothly and faithfully on $C(M)$. Conjecture. It is not possible to have smooth faithful action…

Let $E$ be a smooth vector bundle over a smooth manifold $M$, and let $G$ be a smooth generalized subbundle of $E$. A generalized subbundle is finitely generated over an open set…

A smooth, closed manifold is a smooth manifold without boundary that is compact. A nonsingular algebraic set is a real algebraic set whose singular locus is empty. Nash's conjectur…