  A series of old and recent theoretical observations suggests that the
quantization of gravity would be feasible, and some problems of Quantum Field
Theory would go away if, somehow, the spacetime would undergo a dimensional
reduction at high energy scales. But an identification of the deep mechanism
causing this dimensional reduction would still be desirable. The main
contribution of this article is to show that dimensional reduction effects are
due to General Relativity at singularities, and do not need to be postulated
ad-hoc.
  Recent advances in understanding the geometry of singularities do not require
modification of General Relativity, being just non-singular extensions of its
mathematics to the limit cases. They turn out to work fine for some known types
of cosmological singularities (black holes and FLRW Big-Bang), allowing a
choice of the fundamental geometric invariants and physical quantities which
remain regular. The resulting equations are equivalent to the standard ones
outside the singularities.
  One consequence of this mathematical approach to the singularities in General
Relativity is a special, (geo)metric type of dimensional reduction: at
singularities, the metric tensor becomes degenerate in certain spacetime
directions, and some properties of the fields become independent of those
directions. Effectively, it is like one or more dimensions of spacetime just
vanish at singularities. This suggests that it is worth exploring the
possibility that the geometry of singularities leads naturally to the
spontaneous dimensional reduction needed by Quantum Gravity.
