Sharpness of Rickman's Picard theorem in all dimensions

One of the classical theorems in complex analysis is the
Picard's theorem stating that a non-constant entire holomorphic map from
the complex plane to the Riemann sphere omits at most two points.

In the late 1960's and early 1970's, results of Reshetnyak and
Martio-Rickman-Väisälä showed that mappings of bounded distortion, also
called as quasiregular mappings, can be viewed as a counterpart for
holomorphic mappings in quasiconformal geometry. One of the natural
goals from the very beginning in this theory was obtain Picard-type
results. In 1980, Rickman showed that a non-constant quasiregular
mapping from the Euclidean n-space to the n-sphere omits only finitely
many points, where the number depends only on the dimension and
distortion. The sharpness of Rickman's theorem was not as simple issue
as in the classical Picard theorem. In 1984, Rickman showed using a
surprising and elaborate construction that given any finite set in the
3-sphere there exists a quasiregular from the Euclidean 3-space into the
3-sphere omitting exactly that set.

In this talk, I will discuss joint work with David Drasin on the
sharpness of Rickman's Picard theorem in all dimensions. Especially, I
will discuss the role of bilipschitz geometry in the proof which leads
to a stronger statement on the metric properties of the map and is a
crucial ingredient in dimensions n > 3.