Weekly Bulletin

For any set of algebraic numbers in a fixed number field K satisfying standard metric conditions in the theory (close enough to zero), we prove that the values of Lerch functions (essentially polylogarithms with ``shifts'') are linearly independent over K

A starting point for the general study of embeddings of smooth real manifolds into highly symmetric smooth manifolds were the classical theorems of Whitney and Wu which assert that every smooth real manifold of dimension d≥1 properly embeds into the real Euclidean space of dimension R^(2d) and that two embeddings into R^(max(4,2d+1)) are the same up to diffeomorphisms. In this talk, we first address general embedding theorems for smooth affine algebraic varieties into complex Euclidean space due to Kaliman-Nori-Srinivas and Gromov-Eliashberg. Then we focus on the existence and uniqueness of embeddings of smooth affine varieties into complex linear algebraic groups up to algebraic (holomorphic) automorphisms of the underlying variety. This covers joint work with Peter Feller and Jérémy Blanc.