Descartes proved that a graph embedding into the plane on V vertices and E edges satisfies

                                                                                         E<3

Quite disappointingly, no similarly beautiful numerical inequality exists to limit the number of faces of, say, 2-dimensional simplicial complexes embedding in dimension 4, and topological techniques seem to be limited in their reach to provide us with such a result. I will instead use algebra and combinatorics to generalize Descartes' result.

Presentation by: Karim Adiprasito. Hebrew University of Jerusalem, Israel.