  We consider parametric Feynman integrals and their dimensional regularization
from the point of view of differential forms on hypersurface complements and
the approach to mixed Hodge structures via oscillatory integrals. We consider
restrictions to linear subspaces that slice the singular locus, to handle the
presence of non-isolated singularities. In order to account for all possible
choices of slicing, we encode this extra datum as an enrichment of the Hopf
algebra of Feynman graphs. We introduce a new regularization method for
parametric Feynman integrals, which is based on Leray coboundaries and, like
dimensional regularization, replaces a divergent integral with a Laurent series
in a complex parameter. The Connes--Kreimer formulation of renormalization can
be applied to this regularization method. We relate the dimensional
regularization of the Feynman integral to the Mellin transforms of certain
Gelfand--Leray forms and we show that, upon varying the external momenta, the
Feynman integrals for a given graph span a family of subspaces in the
cohomological Milnor fibration. We show how to pass from regular singular
Picard--Fuchs equations to irregular singular flat equisingular connections. In
the last section, which is more speculative in nature, we propose a geometric
model for dimensional regularization in terms of logarithmic motives and
motivic sheaves.
