A Global Theory of Branched Minimal Immersions

We will discuss a new global approach to  proving that surfaces in three
space which are relative minima for Area are necessarily immersed in the
interior.  After a historical discussion of the problem, we will, for
the first time, provide a strategy enabling one to compute not just the
second variation, but variations of arbitrarily large order of
Dirichlet's Energy defined on the infinite dimensional manifold of
surfaces spanning a curve in three space. Time permitting, we discuss
the potential applications to minimal surfaces of higher genus and to
the discussion of the difficult question of whether or not such minimal
surfaces need be immersed at points on their boundaries.