9 problems

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…

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…

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…

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 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…

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

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