Description
A compact surface is called rigid if the only length-preserving
transformations of it are congruencies of the ambient space - in simple
terms rigid surfaces are unbendable. Rigidity is a topic which is
already
found in Euclid's Elements. Leonhard Euler conjectured in 1766 that
every
smooth compact surface is rigid. The young Augustin Cauchy found a
proof
in 1813 for convex polyhedra, but it took another 100 years until a
proof
for smooth convex surfaces appeared. Whilst it seems clear that
convexity,
and more generally curvature plays an important role for rigidity, it
came
as a shock to the world of geometry when John Nash showed in 1954 that
every
surface can be bent in an essentially arbitrary, albeit non-convex
manner.
The proof of Nash involves a highly intricate fractal-like
construction,
that has in recent years found applications in many different branches
of
applied mathematics, such as the theory of solid-solid phase
transitions
and hydrodynamical turbulence. The talk will provide an overview of
this
fascinating subject and present the most recent developments.