13 problems

The study of surfaces in $\mathbb{R}^4$ from the affine viewpoint is related to affine differential geometry. A convex compact surface is a compact convex surface…

A proper convex cone is a convex cone containing no complete affine line and having nonempty interior. A hyperbolic affine sphere is a hypersurface in affine space whose affine nor…

Let $M \subset \mathbb{R}^4$ be a locally strictly convex surface homeomorphic to the sphere. Loewner's conjecture. The surface $M$ has at least two inflections. This conjecture co…

Let $\mathcal A$ be a complete abelian left-symmetric algebra (LSA) with a nondegenerate Hessian-type inner product. Let $\operatorname{sDer}(\mathcal A)$ denote its space of infin…

A Cayley hypersurface is a hypersurface affinely congruent to the graph of the Cayley polynomial … For a nondegenerate homogeneous hypersurface, the relevant properties are that it…

Cayley hypersurface characterisation conjecture. Then, in a suitable affine coordinate system, $\Sigma$ is given by the Cayley hypersurface equation.

Let $f$ be a smooth strictly convex function defined on a convex domain $\Omega$ in $\mathbb{R}^2$. The Calabi rigidity conjecture. If $f$ satisfies the PDE (1.2) with…

Semi-parallelism and parallelism conjecture. The two conditions are equivalent:

Izumiya's Lagrangian stability conjecture. Germs of generic Blaschke affine normal congruences at any point are Lagrangian stable.

Trudinger–Wang's conjecture. The first boundary value problem has a unique smooth solution $u$ if $\phi$ is affine mean convex. This conjecture concerns existence and uniqueness fo…

Trudinger–Wang's conjecture. If $n\leq 9$, then $u$ is an elliptic quadratic polynomial. If $n\geq 10$, then there exists a smooth, locally uniformly convex solution of the equatio…

Let $M$ be a locally uniformly convex hypersurface in $\mathbb R^n$ that is complete in the equiaffine metric and has equiaffine mean curvature zero. Affine Bernstein conjecture. F…

Let $M\subset\mathbb R^4$ be a locally strictly convex surface, meaning that at each point $p\in M$ there is a tangent hyperplane with non-degenerate contact that locally supports…