The fascinating world of polytopes

A polytope is the convex hull of finitely many points in a Euclidean space. These "simple" objects show up in such diverse areas of Mathematics as Combinatorics and Discrete Geometry, Optimization, Symplectic Geometry, and Algebraic Geometry, to name just a few. While the study of polytopes has been extremely active since the last half of the 20th century, many easy to state problems on face numbers of polytopes remain wide open. In the talk, I'll discuss several such problems as well as touch on some very recent developments.