Abstract: The talk will discuss some recent progress in embeddability of polyhedra in Euclidean spaces and in products of trees. In topology, a polyhedron is a space triangulated by a simplicial complex. The methods involve little to no algebraic topology but a fair amount of combinatorics of cubical complexes and other non-simplicial cell complexes.

A simple consequence is the following "explanation" of the statement of the Kuratowski graph planarity criterion (a graph is planar if and only if it does not "contain" a K₅ nor a K_3,3): There exist precisely two 3-dimensional "dichotomial" cell complexes (that is, cell complexes where to every cell there corresponds a unique cell with the complementary set of vertices), and their 1-skeleta are K₅ and K_3,3.

This is no accident, since 4-dimensional dichotomial cell complexes yield a similar "explanation" of 6/7 of the statement of the famous Robertson–Seymour–Thomas criterion for "intrinsic linking" of graphs in 3-space. Further details can be found in arXiv:1103.5457 and arXiv:1102.0696.

This talk will be accessible to graduate students.