On the monodromy conjecture for non-degenerate hypersurfaces.
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables.
A crucial difference from the case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this
setting. We develop several new tools and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.