4 problems

Let $X=G/P$ and $X'=G'/P'$ be homogeneous projective varieties, where $G$ and $G'$ are connected simple algebraic groups and $P$ and $P'$ are parabolic subgroups. For a realization…

A regular map is a graph embedded in a surface with the regularity property that its automorphism group acts transitively on incident vertex-edge-face flags. A regular map is self-…

Small-genus convex-face conjecture. No regular maps other than the eight currently known examples admit polyhedral embeddings with convex faces.

Grünbaum polyhedron uniqueness conjecture. The Gr"unbaum polyhedron is conjectured to be the only vertex-transitive polyhedron of genus $g\geqslant 2$ that is also combinatorially…