Naum Zobin
Consider the space of functions with bounded k-th derivatives
in a general domain in 
Is every such
function extendable to a function of the same class defined on the whole
? H. Whitney showed in 1934 that the equivalence of the
geodesic metric in this domain to the Euclidean one is
sufficient for such extendability.
There was an old conjecture (going back to H. Whitney) that this equivalence
is also necessary for extendability.
We disprove this conjecture in all dimensions starting from 2.
The counterexamples are infinitely connected domains in 
It is possible to construct counterexamples in 
homeomorphic to balls, so no topological restrictions can help in dimensions
3 and higher. As for dimension 2, we prove that, nevertheless,
the Whitney's Conjecture is true for bounded finitely connected domains.
In this talk we are going to concentrate on explaining the ideas behind the counterexamples in dimension 2 and higher.